Given two points with position vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = -\hat{i} + 4\hat{j} + 2\hat{k}$, find the position vector of the midpoint of the line segment joining these two points.

$\frac{1}{2}(\hat{i} + 7\hat{j} + \hat{k})$

$\frac{1}{2}(\hat{i} + 7\hat{j} - \hat{k})$

$\frac{1}{2}(\hat{i} + 7\hat{j} + 3\hat{k})$

$\frac{1}{2}(3\hat{i} + 7\hat{j} + \hat{k})$

Given two non-zero vectors $\vec{a}$ and $\vec{b}$, if $\vec{a} \times \vec{b} = \vec{0}$, which of the following statements is true?

$\vec{a}$ and $\vec{b}$ are parallel.

$\vec{a}$ and $\vec{b}$ are perpendicular.

$\vec{a}$ and $\vec{b}$ are parallel.

$\vec{a}$ and $\vec{b}$ have the same magnitude.

The angle between $\vec{a}$ and $\vec{b}$ is $90^\circ$.

Given two vectors $\vec{a} = 3\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - 4\hat{j} + 2\hat{k}$, find the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$.

$\frac{1}{\sqrt{29}}(2\hat{i} + \hat{j} - 13\hat{k})$

$\frac{1}{\sqrt{29}}(-2\hat{i} - \hat{j} + 13\hat{k})$

$\frac{1}{\sqrt{29}}(2\hat{i} - \hat{j} - 13\hat{k})$

$\frac{1}{\sqrt{29}}(-2\hat{i} + \hat{j} + 13\hat{k})$

Which of the following statements about the cross product of two vectors $\vec{a}$ and $\vec{b}$ is true?

The cross product $\vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$.

The cross product $\vec{a} \times \vec{b}$ is commutative.

The cross product $\vec{a} \times \vec{b}$ results in a scalar.

The cross product $\vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$.

The magnitude of $\vec{a} \times \vec{b}$ is $|\vec{a}||\vec{b}|\cos\theta$.

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Vector Algebra

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Vector Algebra

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Applications of Vector Products

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Learning Objectives

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Learning Objectives

Understand the distinction between scalar and vector quantities.
Identify and describe basic concepts related to vectors.
Perform operations on vectors, including addition, subtraction, and scalar multiplication.
Calculate the magnitude and direction of vectors.
Apply vector concepts to solve real-world problems involving displacement, velocity, and force.
Analyze the properties of vectors, including collinearity and equality.
Utilize vector algebra in geometric contexts, such as finding areas and angles between vectors.
Explore the applications of vectors in physics and engineering.

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Revision Notes & Summary

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Chapter 10: Vector Algebra

10.1 Introduction

Scalars: Quantities with only magnitude (e.g., height, mass).
Vectors: Quantities with both magnitude and direction (e.g., force, velocity).

10.2 Some Basic Concepts

A directed line can be represented with arrowheads.
Directed Line: A line with a prescribed direction.

Exercises

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
Find the scalar components and magnitude of the vector joining the points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂).
A girl walks 4 km towards west, then 3 km in a direction 30° east of north. Determine her displacement from the initial point.
If ã = b + c, is it true that a = b + c? Justify.
Find the value of x for which x(î + j + k) is a unit vector.
Find a vector of magnitude 5 units, parallel to the resultant of vectors a = 2î + 3ⱼ - k and b = i - 2ⱼ + k.
If a = î + j + k, b = 2î - j + 3k, and c = î - 2ⱼ + k, find a unit vector parallel to 2a - b + 3c.
Show that points A (1, -2, -8), B (5, 0, -2), and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Find the position vector of point R which divides the line joining two points P and Q whose position vectors are (2a + b) and (a - 3b) externally in the ratio 1:2.
The two adjacent sides of a parallelogram are 2î - 4ⱼ + 5k and î - 2ⱼ - 3k. Find the unit vector parallel to its diagonal and its area.
Show that the direction cosines of a vector equally inclined to the axes OX, OY, and OZ are ±(1/√3).
Let b = 3î - 2ⱼ + 7k. Find a vector d which is perpendicular to both a and b, and c.d = 15.
The scalar product of vector i + j + k with a unit vector along the sum of vectors 2î + 4ⱼ - 5k and Ni + 2ⱼ + 3k is equal to one. Find the value of A.
If ã, b, c are mutually perpendicular vectors of equal magnitudes, show that the vector 15 is equally inclined to ã, b, and c.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

It is a scalar.

It is a vector perpendicular to both

\vec{a}

and

\vec{b}

It is a vector parallel to

\vec{a}

It is a vector parallel to

\vec{b}

Correct Answer: B

Solution:

The vector product

\vec{a} \times \vec{b}

results in a vector that is perpendicular to both

\vec{a}

and

\vec{b}

, as defined by the right-hand rule.

Chapter Concept:

Applications of Vector Products

Correct Answer: B

Solution:

The dot product

\vec{a} \cdot \vec{b} = (2)(1) + (3)(-1) + (1)(2) = 2 - 3 + 2 = 1

Chapter Concept:

Vector Algebra

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True or False

Correct Answer: False

Solution:

The projection of a vector on a line is calculated using the dot product, not the cross product.

Chapter Concept :

Projection of a Vector

Correct Answer: True

Solution:

The vector product, also known as the cross product, results in a vector that is perpendicular to the plane containing

\vec{a}

and

\vec{b}

. Its magnitude is

|\vec{a}||\vec{b}|\sin\theta

and the direction is determined by the right-hand rule.

Chapter Concept :

Vector Product (Cross Product)

Correct Answer: True

Solution:

The magnitude of the position vector

\overrightarrow{OP}

is calculated using the distance formula in three-dimensional space, which is

\sqrt{x^2 + y^2 + z^2}

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Vector Algebra

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Applications of Vector Products

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Collinearity of Vectors in Component Form

High

Cross Product in Determinant Form

Learning Objectives

Learning Objectives

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Revision Notes & Summary

Chapter 10: Vector Algebra

10.1 Introduction

10.2 Some Basic Concepts

Exercises

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.Which of the following statements is true about the vector product a⃗×b⃗\vec{a} \times \vec{b}a×b?

Correct Answer: B

Q2.Given vectors a⃗=2i^+3j^+k^\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}a=2i^+3j^​+k^ and b⃗=i^−j^+2k^\vec{b} = \hat{i} - \hat{j} + 2\hat{k}b=i^−j^​+2k^, what is the dot product a⃗⋅b⃗\vec{a} \cdot \vec{b}a⋅b?

Correct Answer: B

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True or False

Q1.The projection of a vector on a line is calculated using the cross product.

Correct Answer: False

Q2.The vector product of two vectors a⃗\vec{a}a and b⃗\vec{b}b is a vector perpendicular to both, with magnitude ∣a⃗∣∣b⃗∣sin⁡θ|\vec{a}||\vec{b}|\sin\theta∣a∣∣b∣sinθ and direction given by the right-hand rule.

Correct Answer: True

Q3.The magnitude of the position vector OP→\overrightarrow{OP}OP for a point P(x,y,z)P(x, y, z)P(x,y,z) is given by x2+y2+z2\sqrt{x^2 + y^2 + z^2}x2+y2+z2​.

Correct Answer: True

High

Cross Product Unit Vector Rules

High

Midpoint Formula Using Position Vectors

High

Perpendicular Vectors Using Dot Product

High

Projection of a Vector

High

Scalars and Vectors

High

Section Formula in Vector Form

High

Types of Vectors

High

Vector Algebra

High

Vector Joining Two Points

High

Vector Multiplication Properties

High

Collinearity of Points Using Vectors

High

Dot Product in Component Form

High

Equal, Collinear and Negative Vectors

High

Parallel Vectors Using Cross Product

High

Parallelogram Law of Vector Addition

High

Scalar Product (Dot Product)

High

Triangle Law of Vector Addition

High

Vector Addition Properties

High

Vector Components

High