Question 1

If a line has direction ratios $a = 3$, $b = 4$, and $c = 5$, what are the direction cosines of the line?

Accepted Answer

Correct Option: A. Solution: Direction cosines $l, m, n$ are given by $l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}$. Substituting $a = 3$, $b = 4$, $c = 5$, we get $l = \frac{3}{\sqrt{50}}, m = \frac{4}{\sqrt{50}}, n = \frac{5}{\sqrt{50}}$.

Question 2

Find the angle between two lines with direction cosines $l_1, m_1, n_1$ and $l_2, m_2, n_2$. Which formula correctly represents this angle?

Accepted Answer

Correct Option: A. Solution: The angle between two lines with direction cosines is given by $\cos \Theta = l_1l_2 + m_1m_2 + n_1n_2$.

Question 3

If the direction cosines of a line are $l = \frac{1}{3}$, $m = \frac{2}{3}$, and $n = \frac{2}{3}$, what is the angle between the line and the $z$-axis?

Accepted Answer

Correct Option: B. Solution: The angle between the line and the $z$-axis is given by $\cos \gamma = n = \frac{2}{3}$. Thus, the angle is $\cos^{-1}\left(\frac{2}{3}\right)$.

Question 4

Given the Cartesian equation of a line $$\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$, which of the following represents the correct vector form of the line?

Accepted Answer

Correct Option: C. Solution: The vector form of a line passing through point $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ is given by $$\mathbf{r} = (x_1, y_1, z_1) + \lambda (ai + bj + ck)$$.

Question 5

Find the direction cosines of a line that makes angles $90^\circ$, $60^\circ$, and $30^\circ$ with the positive x, y, and z-axes, respectively.

Accepted Answer

Correct Option: A. Solution: The direction cosines are given by the cosines of the angles with the axes: $\cos 90^\circ = 0$, $\cos 60^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$.

Question 6

Which mathematical operation is used to find the shortest distance between two skew lines?

Accepted Answer

Correct Option: C. Solution: The cross product of direction vectors is used to find the shortest distance between two skew lines.

Question 7

If a line passes through the point with position vector $\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}$ and is parallel to the vector $\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}$, which of the following points lies on the line?

Accepted Answer

Correct Option: B. Solution: The vector equation of the line is $\vec{r} = \vec{a} + \lambda \vec{b}$. Substituting $\vec{a}$ and $\vec{b}$, we get $\vec{r} = (1 + 2\lambda)\hat{i} + (-2 + 3\lambda)\hat{j} + (4 - \lambda)\hat{k}$. For the point $(5, 4, 1)$, solving $1 + 2\lambda = 5$, $-2 + 3\lambda = 4$, and $4 - \lambda = 1$ gives $\lambda = 2$. Thus, the point lies on the line.

Question 8

A line has direction ratios $3, 4, 5$. Which of the following are the correct direction cosines for this line?

Accepted Answer

Correct Option: A. Solution: The direction cosines are obtained by dividing each direction ratio by the square root of the sum of the squares of the direction ratios: $\sqrt{3^2 + 4^2 + 5^2} = \sqrt{50}$.

Question 9

What is the vector equation of a line passing through two points with position vectors $\mathbf{a}$ and $\mathbf{b}$?

Accepted Answer

Correct Option: A. Solution: The vector equation of a line passing through two points with position vectors $\mathbf{a}$ and $\mathbf{b}$ is given by $\mathbf{r} = \mathbf{a} + \lambda(\mathbf{b} - \mathbf{a})$.

Question 10

Given the direction ratios $a = 3$, $b = 4$, $c = 5$, what are the direction cosines?

Accepted Answer

Correct Option: A. Solution: The direction cosines are calculated as $\frac{a}{\sqrt{a^2 + b^2 + c^2}}, \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \frac{c}{\sqrt{a^2 + b^2 + c^2}}$. Substituting $a = 3$, $b = 4$, $c = 5$, we get $\frac{3}{\sqrt{50}}, \frac{4}{\sqrt{50}}, \frac{5}{\sqrt{50}}$.

Question 11

What is the angle between two lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ given by the formula $$\cos \Theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$$?

Accepted Answer

Correct Option: A. Solution: The angle between two lines is $0^\circ$ if they are parallel, which means their direction ratios are proportional.

Question 12

Determine the Cartesian equation of a line that passes through the point $(2, -1, 3)$ and is parallel to the vector $\mathbf{b} = \begin{bmatrix} 1 \ -2 \ 2 \end{bmatrix}$.

Accepted Answer

Correct Option: A. Solution: The direction ratios of the line are given by the vector $\mathbf{b} = \begin{bmatrix} 1 \ -2 \ 2 \end{bmatrix}$, so the Cartesian equation is $\frac{x-2}{1} = \frac{y+1}{-2} = \frac{z-3}{2}$.

Question 13

If a line makes angles $\alpha = 60^\circ$, $\beta = 45^\circ$, and $\gamma = 60^\circ$ with the coordinate axes, what are its direction cosines?

Accepted Answer

Correct Option: A. Solution: The direction cosines are $l = \cos 60^\circ = \frac{1}{2}$, $m = \cos 45^\circ = \frac{1}{\sqrt{2}}$, $n = \cos 60^\circ = \frac{1}{2}$.

Question 14

Given two skew lines with direction vectors $\mathbf{b}_1 = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}$ and $\mathbf{b}_2 = -\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$, what is the shortest distance between them if the lines pass through points $\mathbf{a}_1 = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ and $\mathbf{a}_2 = 2\mathbf{i} - \mathbf{j} + \mathbf{k}$ respectively?

Accepted Answer

Correct Option: A. Solution: The shortest distance $d$ between two skew lines is given by $d = \frac{|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 	imes \mathbf{b}_2)|}{|\mathbf{b}_1 	imes \mathbf{b}_2|}$. Calculating $\mathbf{b}_1 	imes \mathbf{b}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 3 & -1 \ -1 & 4 & 2 \end{vmatrix} = 10\mathbf{i} + 5\mathbf{j} + 11\mathbf{k}$ and $\mathbf{a}_2 - \mathbf{a}_1 = \mathbf{i} - 3\mathbf{j} - 2\mathbf{k}$, we find $|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 	imes \mathbf{b}_2)| = 26$ and $|\mathbf{b}_1 	imes \mathbf{b}_2| = \sqrt{10^2 + 5^2 + 11^2} = \sqrt{246}$. Therefore, $d = \frac{26}{\sqrt{246}} = \frac{\sqrt{26}}{3}$.

Question 15

Given two points $A(2, -1, 3)$ and $B(5, 4, -2)$, find the vector equation of the line passing through these points.

Accepted Answer

Correct Option: A. Solution: The direction vector is obtained by subtracting the coordinates of $A$ from $B$: $(5-2)\mathbf{i} + (4+1)\mathbf{j} + (-2-3)\mathbf{k} = 3\mathbf{i} + 5\mathbf{j} - 5\mathbf{k}$. Therefore, the vector equation is $\mathbf{r} = (2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) + \lambda(3\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})$.

Question 16

A line in three-dimensional space is perpendicular to two given lines with direction vectors $\mathbf{b_1} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$ and $\mathbf{b_2} = \begin{bmatrix} 4 \ 5 \ 6 \end{bmatrix}$. What is the direction vector of the line that is perpendicular to both given lines?

Accepted Answer

Correct Option: A. Solution: The direction vector of the line perpendicular to both given lines is the cross product of their direction vectors: $$\mathbf{b_1} 	imes \mathbf{b_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 2 & 3 \ 4 & 5 & 6 \end{vmatrix} = \hat{i}(2 \cdot 6 - 3 \cdot 5) - \hat{j}(1 \cdot 6 - 3 \cdot 4) + \hat{k}(1 \cdot 5 - 2 \cdot 4) = \begin{bmatrix} -3 \ 6 \ -3 \end{bmatrix}$$.

Question 17

If the direction cosines of a line are $l, m, n$, which of the following equations must hold true?

Accepted Answer

Correct Option: A. Solution: The direction cosines of a line satisfy the equation $l^2 + m^2 + n^2 = 1$.

Question 18

If a directed line makes angles $\alpha$, $\beta$, and $\gamma$ with the positive $x$, $y$, and $z$ axes respectively, what happens to these angles and the direction cosines if the direction of the line is reversed?

Accepted Answer

Correct Option: B. Solution: When the direction of a line is reversed, the direction angles become their supplements, i.e., $\pi - \alpha$, $\pi - \beta$, and $\pi - \gamma$, and the signs of the direction cosines are reversed.

Question 19

Consider a line in 3D space given by the parametric equations $x = 2 + \lambda$, $y = 3 + 2\lambda$, $z = -1 + 3\lambda$. What is the Cartesian equation of this line?

Accepted Answer

Correct Option: A. Solution: The parametric equations $x = 2 + \lambda$, $y = 3 + 2\lambda$, $z = -1 + 3\lambda$ correspond to the direction ratios $1, 2, 3$. The Cartesian form is $\frac{x - 2}{1} = \frac{y - 3}{2} = \frac{z + 1}{3}$.

Question 20

Given the Cartesian line equation $$\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$, what is the vector form of the line?

Accepted Answer

Correct Option: A. Solution: The vector form of the line is $$\mathbf{r} = \mathbf{a}_0 + \lambda \mathbf{b}$$ where $$\mathbf{a}_0$$ is the position vector of the point $$(x_1, y_1, z_1)$$ and $$\mathbf{b}$$ is the direction vector $$ai + bj + ck$$.

Question 21

A line makes angles $\alpha$, $\beta$, and $\gamma$ with the $x$, $y$, and $z$ axes respectively. If $\cos \alpha = \frac{3}{5}$ and $\cos \beta = \frac{4}{5}$, what is $\cos \gamma$?

Accepted Answer

Correct Option: A. Solution: Using the identity $l^2 + m^2 + n^2 = 1$, where $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$, we have $\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 + n^2 = 1$. Solving gives $n = \frac{1}{5}$.

Question 22

What is the formula for calculating the shortest distance between two skew lines given by their vector equations $$\mathbf{r} = \mathbf{a}_1 + \lambda \mathbf{b}_1$$ and $$\mathbf{r} = \mathbf{a}_2 + \mu \mathbf{b}_2$$?

Accepted Answer

Correct Option: A. Solution: The shortest distance between two skew lines is given by the formula $$\frac{|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 	imes \mathbf{b}_2)|}{|\mathbf{b}_1 	imes \mathbf{b}_2|}$$.

Question 23

If a line passes through the points $P(1, 2, 3)$ and $Q(4, 6, 8)$, what are the direction ratios of the line?

Accepted Answer

Correct Option: B. Solution: The direction ratios of the line passing through points $P(1, 2, 3)$ and $Q(4, 6, 8)$ are given by $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$, which are $(4-1, 6-2, 8-3) = (3, 4, 5)$.

Question 24

Which of the following statements is true about skew lines?

Accepted Answer

Correct Option: C. Solution: Skew lines are defined as lines in space that are neither parallel nor intersecting. They lie in different planes, making them non-coplanar.

Question 25

If two lines have direction ratios $a_1 = 1, b_1 = -2, c_1 = 2$ and $a_2 = 2, b_2 = 1, c_2 = -1$, what is the shortest distance between them if they pass through points $P(1, 0, 0)$ and $Q(0, 1, 1)$ respectively?

Accepted Answer

Correct Option: A. Solution: The shortest distance $d$ between two skew lines is given by $d = \frac{|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 	imes \mathbf{b}_2)|}{|\mathbf{b}_1 	imes \mathbf{b}_2|}$. Here, $\mathbf{b}_1 = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ and $\mathbf{b}_2 = 2\mathbf{i} + \mathbf{j} - \mathbf{k}$. Calculating $\mathbf{b}_1 	imes \mathbf{b}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & -2 & 2 \ 2 & 1 & -1 \end{vmatrix} = 0\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}$ and $\mathbf{a}_2 - \mathbf{a}_1 = -\mathbf{i} + \mathbf{j} + \mathbf{k}$, we find $|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{b}_1 	imes \mathbf{b}_2)| = 10$ and $|\mathbf{b}_1 	imes \mathbf{b}_2| = \sqrt{50}$. Therefore, $d = \frac{10}{\sqrt{50}} = \frac{3}{\sqrt{14}}$.

Question 26

Determine the angle between the lines with direction ratios $1, 2, 3$ and $2, 3, 4$.

Accepted Answer

Correct Option: A. Solution: The angle between two lines is given by $\cos \Theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}}$. Substituting the given direction ratios, we find $\cos \Theta = \frac{1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4}{\sqrt{1^2 + 2^2 + 3^2} \cdot \sqrt{2^2 + 3^2 + 4^2}} = \frac{20}{\sqrt{14} \cdot \sqrt{29}}$.

Question 27

A line is perpendicular to two given lines with direction vectors $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}$ and $\mathbf{b} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$. What is the direction vector of the line perpendicular to both?

Accepted Answer

Correct Option: B. Solution: The direction vector of the line perpendicular to both given lines is parallel to the cross product of their direction vectors: $\mathbf{a} 	imes \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 3 & 1 \ 1 & -2 & 2 \end{vmatrix} = \mathbf{i}(-6 - 1) - \mathbf{j}(4 - 1) + \mathbf{k}(-4 - 3) = -7\mathbf{i} - 3\mathbf{j} - 7\mathbf{k}$. Simplifying gives the direction vector $\mathbf{i} - 5\mathbf{j} + 7\mathbf{k}$.

Question 28

A line has direction ratios $a = 1$, $b = 2$, $c = 2$. If the line is reversed, what will be the new direction cosines?

Accepted Answer

Correct Option: C. Solution: Reversing the line changes the signs of the direction cosines. The direction cosines are $l = \frac{1}{\sqrt{9}}$, $m = \frac{2}{\sqrt{9}}$, $n = \frac{2}{\sqrt{9}}$. Therefore, the new direction cosines are $\pm \frac{1}{\sqrt{6}}, \pm \frac{2}{\sqrt{6}}, \pm \frac{2}{\sqrt{6}}$.

Question 29

What are the direction cosines of the z-axis?

Accepted Answer

Correct Option: C. Solution: The direction cosines of the z-axis are (0, 0, 1) because it makes an angle of 0° with the z-axis and 90° with the x and y-axes.

Question 30

For a line with direction cosines $l, m, n$, if $l = \frac{1}{2}$, $m = \frac{1}{\sqrt{2}}$, what is $n$?

Accepted Answer

Correct Option: A. Solution: Using $l^2 + m^2 + n^2 = 1$, we have $\left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + n^2 = 1$. Solving gives $n = \frac{1}{2}$.

Question 31

If two lines in space are neither parallel nor intersecting, what are they called?

Accepted Answer

Correct Option: B. Solution: Lines that are neither parallel nor intersecting are called skew lines.

Question 32

Which of the following represents the Cartesian equation of a line through a point $(x_1, y_1, z_1)$ with direction ratios $a, b, c$?

Accepted Answer

Correct Option: A. Solution: The Cartesian equation of a line through a point $(x_1, y_1, z_1)$ with direction ratios $a, b, c$ is given by $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$.

Question 33

Given two lines in space with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$, which of the following expressions correctly represents the condition for these lines to be parallel?

Accepted Answer

Correct Option: B. Solution: Two lines are parallel if their direction ratios are proportional, i.e., $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

Question 34

Consider three points $A(1, 2, 3)$, $B(4, 5, 6)$, and $C(7, 8, 9)$ in 3D space. Are these points collinear?

Accepted Answer

Correct Option: A. Solution: The direction ratios of $AB$ are $(4-1, 5-2, 6-3) = (3, 3, 3)$ and those of $BC$ are $(7-4, 8-5, 9-6) = (3, 3, 3)$. Since they are proportional, the points are collinear.

Question 35

Given two points $A(1, 2, 3)$ and $B(4, 6, 8)$, find the direction cosines of the line passing through these points.

Accepted Answer

Correct Option: A. Solution: The direction ratios of the line are $(4-1, 6-2, 8-3) = (3, 4, 5)$. The magnitude is $\sqrt{3^2 + 4^2 + 5^2} = \sqrt{50}$. Thus, the direction cosines are $\frac{3}{\sqrt{50}}, \frac{4}{\sqrt{50}}, \frac{5}{\sqrt{50}}$.

Question 36

A line in space has direction ratios $2, -1, 2$. If the line passes through the point $(3, -2, 5)$, what is the vector equation of the line?

Accepted Answer

Correct Option: A. Solution: The vector equation of a line with direction ratios $2, -1, 2$ passing through $(3, -2, 5)$ is $\mathbf{r} = (3, -2, 5) + \lambda (2, -1, 2)$.

Question 37

Two lines in 3D space with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are perpendicular if which of the following conditions is satisfied?

Accepted Answer

Correct Option: A. Solution: Two lines are perpendicular if the dot product of their direction ratios is zero, i.e., $a_1a_2 + b_1b_2 + c_1c_2 = 0$.

Question 38

A directed line in space makes angles $\alpha$, $\beta$, and $\gamma$ with the positive $x$, $y$, and $z$ axes, respectively. If the direction of the line is reversed, what will be the new direction angles with the axes?

Accepted Answer

Correct Option: A. Solution: Reversing the direction of a line changes its direction angles to their supplements, i.e., $\pi - \alpha$, $\pi - \beta$, and $\pi - \gamma$.

Question 39

If a line passes through the point $(2, -1, 3)$ and has direction ratios $4, -2, 5$, what is the Cartesian equation of the line?

Accepted Answer

Correct Option: A. Solution: The Cartesian equation of a line through $(2, -1, 3)$ with direction ratios $4, -2, 5$ is $$\frac{x-2}{4} = \frac{y+1}{-2} = \frac{z-3}{5}$$.

Question 40

Which of the following statements correctly describes skew lines?

Accepted Answer

Correct Option: C. Solution: Skew lines are defined as lines in space that are neither parallel nor intersecting, and they are non-coplanar.

Question 41

If a line passes through the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, what is the expression for the direction cosines of the line?

Accepted Answer

Correct Option: A. Solution: The direction cosines of the line passing through points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are given by $\frac{x_2 - x_1}{PQ}, \frac{y_2 - y_1}{PQ}, \frac{z_2 - z_1}{PQ}$ where $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

Question 42

A line passes through the points $(1, 2, 3)$ and $(4, 6, 8)$. What are the direction cosines of this line?

Accepted Answer

Correct Option: D. Solution: The direction ratios are $(4-1, 6-2, 8-3) = (3, 4, 5)$. The direction cosines are $\left(\frac{3}{\sqrt{3^2 + 4^2 + 5^2}}, \frac{4}{\sqrt{3^2 + 4^2 + 5^2}}, \frac{5}{\sqrt{3^2 + 4^2 + 5^2}}\right) = \left(\frac{3}{\sqrt{50}}, \frac{4}{\sqrt{50}}, \frac{5}{\sqrt{50}}\right)$.

Question 43

What are the direction ratios of a line segment joining two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$?

Accepted Answer

Correct Option: A. Solution: The direction ratios of the line segment joining $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are given by $x_2 - x_1, y_2 - y_1, z_2 - z_1$.

Question 44

If a line passes through the point $A(1, 2, 3)$ and is parallel to the vector $\mathbf{b} = 3\mathbf{i} + 2\mathbf{j} - 8\mathbf{k}$, what is the Cartesian equation of the line?

Accepted Answer

Correct Option: A. Solution: The Cartesian equation of a line through the point $(1, 2, 3)$ and parallel to the vector $3\mathbf{i} + 2\mathbf{j} - 8\mathbf{k}$ is $\frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{-8}$.

Question 45

How do you find the direction cosines of a line passing through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$?

Accepted Answer

Correct Option: A. Solution: The direction cosines of the line segment joining the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are obtained by dividing the direction ratios $x_2 - x_1, y_2 - y_1, z_2 - z_1$ by the distance between the points, $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

Question 46

What is the relationship between direction cosines $l, m, n$ and direction ratios $a, b, c$ of a line?

Accepted Answer

Correct Option: A. Solution: Direction cosines are proportional to direction ratios, so $l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}$.

Question 47

Given two points $P(1, 2, 3)$ and $Q(4, 6, 8)$, what are the direction cosines of the line segment joining these points?

Accepted Answer

Correct Option: A. Solution: The direction ratios are $3, 4, 5$ and the magnitude is $\sqrt{3^2 + 4^2 + 5^2} = \sqrt{50}$. Thus, the direction cosines are $\frac{3}{\sqrt{50}}, \frac{4}{\sqrt{50}}, \frac{5}{\sqrt{50}}$.

Question 48

For the lines $l_1: \frac{x-1}{2} = \frac{y+3}{-1} = \frac{z-4}{1}$ and $l_2: \frac{x-2}{3} = \frac{y-1}{2} = \frac{z+1}{-2}$, determine if they are parallel, intersecting, or skew.

Accepted Answer

Correct Option: C. Solution: The direction vectors for $l_1$ and $l_2$ are $\mathbf{b}_1 = 2\mathbf{i} - \mathbf{j} + \mathbf{k}$ and $\mathbf{b}_2 = 3\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}$, respectively. The lines are skew if they are neither parallel nor intersecting. Since $\mathbf{b}_1$ is not a scalar multiple of $\mathbf{b}_2$, they are not parallel. To check for intersection, solve the system of equations derived from the line equations for a common point, which does not exist here, confirming they are skew.

Question 49

Find the Cartesian equation of a line passing through the point $(1, -2, 3)$ and parallel to the vector $4\mathbf{i} - \mathbf{j} + 2\mathbf{k}$.

Accepted Answer

Correct Option: A. Solution: The Cartesian form of the line is given by $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$ where $(x_1, y_1, z_1)$ is a point on the line and $(a, b, c)$ are the direction ratios. Substituting the given values, we get $\frac{x - 1}{4} = \frac{y + 2}{-1} = \frac{z - 3}{2}$.

Question 50

What is the shortest distance between two skew lines in three-dimensional space?

Accepted Answer

Correct Option: A. Solution: The shortest distance between two skew lines is along the common perpendicular to both lines.

Question 51

Given the points $P(2, -1, 4)$, $Q(3, 0, 5)$, and $R(4, 1, 6)$, determine if they are collinear.

Accepted Answer

Correct Option: A. Solution: The direction ratios of $PQ$ are $(3-2, 0+1, 5-4) = (1, 1, 1)$ and those of $QR$ are $(4-3, 1-0, 6-5) = (1, 1, 1)$. Since they are proportional, the points are collinear.

Question 52

Consider a line passing through the point $(2, -3, 5)$ with direction ratios $4, -1, 3$. Which of the following is the correct Cartesian equation of the line?

Accepted Answer

Correct Option: A. Solution: The Cartesian equation of a line through a point $(x_1, y_1, z_1)$ with direction ratios $a, b, c$ is given by $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$. Substituting the given values, we get $\frac{x-2}{4} = \frac{y+3}{-1} = \frac{z-5}{3}$.

Question 53

Consider a line that passes through the origin and makes angles $\alpha$, $\beta$, and $\gamma$ with the $x$, $y$, and $z$-axes, respectively. If the direction cosines of this line are given by $l = \cos \alpha$, $m = \cos \beta$, and $n = \cos \gamma$, which of the following conditions must be true for the line to align with the $z$-axis?

Accepted Answer

Correct Option: B. Solution: For a line to align with the $z$-axis, it must have direction cosines $l = 0$, $m = 0$, and $n = 1$. This is because the line makes an angle of $0^\circ$ with the $z$-axis and $90^\circ$ with both the $x$ and $y$-axes, resulting in $\cos 0^\circ = 1$ and $\cos 90^\circ = 0$.

Question 54

If a line is given by the Cartesian equation $$\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$, identify the direction vector.

Accepted Answer

Correct Option: A. Solution: The direction vector of the line is $$ai + bj + ck$$, derived directly from the coefficients of the parameters in the Cartesian equation.

Question 55

If a line makes angles $90^\circ$, $60^\circ$, and $30^\circ$ with the positive direction of the $x$, $y$, and $z$-axes respectively, what are its direction cosines?

Accepted Answer

Correct Option: A. Solution: The direction cosines are given by $l = \cos 90^\circ = 0$, $m = \cos 60^\circ = \frac{1}{2}$, and $n = \cos 30^\circ = \frac{\sqrt{3}}{2}$.

Question 56

Find the direction cosines of the line passing through the points $(-2, 4, -5)$ and $(1, 2, 3)$.

Accepted Answer

Correct Option: A. Solution: The direction cosines are calculated as $\frac{x_2 - x_1}{PQ}, \frac{y_2 - y_1}{PQ}, \frac{z_2 - z_1}{PQ}$ where $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} = \sqrt{77}$. Thus, the direction cosines are $\frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, \frac{8}{\sqrt{77}}$.

Question 57

What is the vector equation of a line that passes through a point with position vector $\mathbf{a}$ and is parallel to a vector $\mathbf{b}$?

Accepted Answer

Correct Option: A. Solution: The vector equation of a line through a point with position vector $\mathbf{a}$ and parallel to vector $\mathbf{b}$ is given by $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$.

Question 58

What is the condition for two lines to be parallel in terms of their direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$?

Accepted Answer

Correct Option: A. Solution: Two lines are parallel if their direction ratios are proportional, i.e., $a_1/a_2 = b_1/b_2 = c_1/c_2$.

Question 59

Consider the line equation $$\frac{x-1}{2} = \frac{y+3}{-1} = \frac{z-4}{3}$$. Which of the following points lies on this line?

Accepted Answer

Correct Option: A. Solution: Substituting $(3, 1, 10)$ into $$\frac{x-1}{2} = \frac{y+3}{-1} = \frac{z-4}{3}$$ gives $$\frac{3-1}{2} = 1$$, $$\frac{1+3}{-1} = -4$$, and $$\frac{10-4}{3} = 2$$. The values do not match, so the point does not lie on the line. Correct point should satisfy all ratios.

Question 60

If a line has direction ratios $2, -1, -2$, what are its direction cosines?

Accepted Answer

Correct Option: A. Solution: The direction cosines are calculated as $\frac{a}{\sqrt{a^2 + b^2 + c^2}}, \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \frac{c}{\sqrt{a^2 + b^2 + c^2}}$. Given $a = 2$, $b = -1$, $c = -2$, we find $\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3}$.

Question 61

For points $X(0, 0, 0)$, $Y(1, 1, 1)$, and $Z(2, 2, 2)$, are these points collinear in 3D space?

Accepted Answer

Correct Option: A. Solution: The direction ratios of $XY$ are $(1-0, 1-0, 1-0) = (1, 1, 1)$ and those of $YZ$ are $(2-1, 2-1, 2-1) = (1, 1, 1)$. Since they are proportional, the points are collinear.

Question 62

A line has direction cosines $l, m, n$ such that $l^2 + m^2 + n^2 = 1$. If the line passes through the point $(1, 2, 3)$ and is parallel to the vector $3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$, what are the direction cosines of the line?

Accepted Answer

Correct Option: A. Solution: The direction cosines $l, m, n$ are proportional to the direction ratios $3, 4, 5$. Therefore, $l = \frac{3}{\sqrt{3^2 + 4^2 + 5^2}}, m = \frac{4}{\sqrt{3^2 + 4^2 + 5^2}}, n = \frac{5}{\sqrt{3^2 + 4^2 + 5^2}}$. Calculating, we get $l = \frac{3}{\sqrt{50}}, m = \frac{4}{\sqrt{50}}, n = \frac{5}{\sqrt{50}}$.

Question 63

What is the shortest distance between the lines given by $\mathbf{r} = (1, 2, 3) + \lambda(2, -1, 4)$ and $\mathbf{r} = (4, -1, 2) + \mu(1, 3, -2)$?

Accepted Answer

Correct Option: A. Solution: The shortest distance between two skew lines is given by $\frac{|\mathbf{b_1} 	imes \mathbf{b_2}|}{|\mathbf{b_1}|}$, where $\mathbf{b_1}$ and $\mathbf{b_2}$ are the direction vectors of the lines. Here, $\mathbf{b_1} = (2, -1, 4)$ and $\mathbf{b_2} = (1, 3, -2)$. The cross product $\mathbf{b_1} 	imes \mathbf{b_2}$ is calculated and its magnitude is divided by $\sqrt{14}$, the magnitude of $\mathbf{b_1}$.

Question 64

Given two parallel lines with equations $$r = \mathbf{a}_1 + \lambda \mathbf{b}$$ and $$r = \mathbf{a}_2 + \mu \mathbf{b}$$, what is the formula for the distance between these lines?

Accepted Answer

Correct Option: A. Solution: The distance between two parallel lines is given by the formula $$\frac{|\mathbf{b} \times (\mathbf{a}_2 - \mathbf{a}_1)|}{|\mathbf{b}|}$$, where $\mathbf{b}$ is the direction vector common to both lines.

Three Dimensional Geometr..

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter 11: Three Dimensional Geometry

11.1 Introduction

11.2 Direction Cosines and Direction Ratios of a Line

11.3 Equation of a Line in Space

11.5 Shortest Distance between Two Lines

Important Notes

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Three Dimensional Geometry

Common Pitfalls

Tips for Success

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Multiple Choice Questions

Q1.If a line has direction ratios a=3a = 3a=3, b=4b = 4b=4, and c=5c = 5c=5, what are the direction cosines of the line?

Correct Answer: A

Q2.Find the angle between two lines with direction cosines l1,m1,n1l_1, m_1, n_1l1​,m1​,n1​ and l2,m2,n2l_2, m_2, n_2l2​,m2​,n2​. Which formula correctly represents this angle?

Correct Answer: A

Q3.If the direction cosines of a line are l=13l = \frac{1}{3}l=31​, m=23m = \frac{2}{3}m=32​, and n=23n = \frac{2}{3}n=32​, what is the angle between the line and the zzz-axis?

Correct Answer: B

Q4.Given the Cartesian equation of a line x−x1a=y−y1b=z−z1c\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}ax−x1​​=by−y1​​=cz−z1​​, which of the following represents the correct vector form of the line?

Correct Answer: C

Q5.Find the direction cosines of a line that makes angles 90∘90^\circ90∘, 60∘60^\circ60∘, and 30∘30^\circ30∘ with the positive x, y, and z-axes, respectively.

Correct Answer: A

Q6.Which mathematical operation is used to find the shortest distance between two skew lines?

Correct Answer: C

Q7.If a line passes through the point with position vector a⃗=i^−2j^+4k^\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}a=i^−2j^​+4k^ and is parallel to the vector b⃗=2i^+3j^−k^\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}b=2i^+3j^​−k^, which of the following points lies on the line?

Correct Answer: B

Q8.A line has direction ratios 3,4,53, 4, 53,4,5. Which of the following are the correct direction cosines for this line?

Correct Answer: A

Q9.What is the vector equation of a line passing through two points with position vectors a\mathbf{a}a and b\mathbf{b}b?

Correct Answer: A

Q10.Given the direction ratios a=3a = 3a=3, b=4b = 4b=4, c=5c = 5c=5, what are the direction cosines?

Correct Answer: A

Correct Answer: A

Q12.Determine the Cartesian equation of a line that passes through the point (2,−1,3)(2, -1, 3)(2,−1,3) and is parallel to the vector b=[1−22]\mathbf{b} = \begin{bmatrix} 1 \\ -2 \\ 2 \end{bmatrix}b=​1−22​​.

Correct Answer: A

Q13.If a line makes angles α=60∘\alpha = 60^\circα=60∘, β=45∘\beta = 45^\circβ=45∘, and γ=60∘\gamma = 60^\circγ=60∘ with the coordinate axes, what are its direction cosines?

Correct Answer: A

Correct Answer: A

Q15.Given two points A(2,−1,3)A(2, -1, 3)A(2,−1,3) and B(5,4,−2)B(5, 4, -2)B(5,4,−2), find the vector equation of the line passing through these points.

Correct Answer: A

Correct Answer: A

Q17.If the direction cosines of a line are l,m,nl, m, nl,m,n, which of the following equations must hold true?

Correct Answer: A

Q18.If a directed line makes angles α\alphaα, β\betaβ, and γ\gammaγ with the positive xxx, yyy, and zzz axes respectively, what happens to these angles and the direction cosines if the direction of the line is reversed?

Correct Answer: B

Q19.Consider a line in 3D space given by the parametric equations x=2+λx = 2 + \lambdax=2+λ, y=3+2λy = 3 + 2\lambday=3+2λ, z=−1+3λz = -1 + 3\lambdaz=−1+3λ. What is the Cartesian equation of this line?

Correct Answer: A

Q20.Given the Cartesian line equation x−x1a=y−y1b=z−z1c\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}ax−x1​​=by−y1​​=cz−z1​​, what is the vector form of the line?

Correct Answer: A

Q21.A line makes angles α\alphaα, β\betaβ, and γ\gammaγ with the xxx, yyy, and zzz axes respectively. If cos⁡α=35\cos \alpha = \frac{3}{5}cosα=53​ and cos⁡β=45\cos \beta = \frac{4}{5}cosβ=54​, what is cos⁡γ\cos \gammacosγ?

Correct Answer: A

Q22.What is the formula for calculating the shortest distance between two skew lines given by their vector equations r=a1+λb1\mathbf{r} = \mathbf{a}_1 + \lambda \mathbf{b}_1r=a1​+λb1​ and r=a2+μb2\mathbf{r} = \mathbf{a}_2 + \mu \mathbf{b}_2r=a2​+μb2​?

Correct Answer: A

Q23.If a line passes through the points P(1,2,3)P(1, 2, 3)P(1,2,3) and Q(4,6,8)Q(4, 6, 8)Q(4,6,8), what are the direction ratios of the line?

Correct Answer: B

Q24.Which of the following statements is true about skew lines?

Correct Answer: C

Correct Answer: A

Q26.Determine the angle between the lines with direction ratios 1,2,31, 2, 31,2,3 and 2,3,42, 3, 42,3,4.

Correct Answer: A

Q27.A line is perpendicular to two given lines with direction vectors a=2i+3j+k\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}a=2i+3j+k and b=i−2j+2k\mathbf{b} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}b=i−2j+2k. What is the direction vector of the line perpendicular to both?

Correct Answer: B

Q28.A line has direction ratios a=1a = 1a=1, b=2b = 2b=2, c=2c = 2c=2. If the line is reversed, what will be the new direction cosines?

Correct Answer: C

Q29.What are the direction cosines of the z-axis?

Correct Answer: C

Q30.For a line with direction cosines l,m,nl, m, nl,m,n, if l=12l = \frac{1}{2}l=21​, m=12m = \frac{1}{\sqrt{2}}m=2​1​, what is nnn?

Correct Answer: A

Q31.If two lines in space are neither parallel nor intersecting, what are they called?

Correct Answer: B