Are these practice questions for Vector Algebra based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Vector Algebra free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Vector Algebra mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Vector Algebra for full coverage.

How many practice questions are there for Vector Algebra?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Vector Algebra Class 12 Mathematics Practice Questions with Answers

easy

Define a 'zero vector' and state one of its properties regarding direction.

Try solving it in your notebook first, then check the solution.

easy

State the two conditions that must be met for two vectors

\vec{a}

and

\vec{b}

to be considered equal.

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easy

Formulate a condition using the dot product for three non-zero vectors

\vec{a}

,

\vec{b}

, and

\vec{c}

to represent the sides of a right-angled triangle.

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easy

If

\vec{a} = \hat{i} - 2\hat{j}

and

\vec{b} = 3\hat{i} + \hat{j}

, calculate the magnitude of the vector

\vec{a} + \vec{b}

.

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easy

Calculate the projection of the vector

\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}

on the vector

\vec{b} = 2\hat{i} + 2\hat{j} + \hat{k}

.

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easy

Calculate the unit vector in the direction of the vector

\vec{v} = 4\hat{i} - 3\hat{j} + 12\hat{k}

.

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easy

Explain the difference between a scalar and a vector quantity. Provide two examples of each from physics.

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easy

Derive the formula for the projection of vector

\vec{a}

onto a non-zero vector

\vec{b}

and justify each step of the derivation.

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easy

State the formula for the scalar product of two vectors

\vec{a}

and

\vec{b}

given in their component forms,

\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}

and

\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}

.

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easy

What are 'coinitial vectors'? Illustrate with a simple sketch.

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medium

Prove that for any triangle ABC, the area can be formulated as

\frac{1}{2} |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|

, where

\vec{a}

,

\vec{b}

, and

\vec{c}

are the position vectors of the vertices A, B, and C respectively.

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medium

Calculate the area of the triangle with vertices A(1, 2, 0), B(2, 0, 2), and C(3, 3, 1).

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medium

Design a non-zero vector

\vec{d}

that is perpendicular to both

\vec{a} = \hat{i} - \hat{j} + \hat{k}

and

\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}

, and satisfies the condition

\vec{c} \cdot \vec{d} = 1

, where

\vec{c} = \hat{i} + 2\hat{j} + 3\hat{k}

.

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medium

List and describe five fundamental types of vectors: Zero Vector, Unit Vector, Collinear Vectors, Equal Vectors, and Negative of a Vector.

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medium

Analyze if the vectors

\vec{p} = 2\hat{i} - 3\hat{j} + \hat{k}

and

\vec{q} = -6\hat{i} + 9\hat{j} - 3\hat{k}

are collinear.

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medium

A student claims that if

|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|

, then vectors

\vec{a}

and

\vec{b}

must be zero vectors. Critique this statement and provide the correct conclusion.

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medium

Describe the Triangle Law of Vector Addition. Use a diagram to explain how the sum of two vectors is represented. What is the resultant vector when the three sides of a triangle are taken in order?

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medium

Calculate the angle between the vectors

\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}

and

\vec{b} = \hat{i} - 2\hat{j} + 2\hat{k}

.

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medium

Analyze if the points A(2, -1, 1), B(1, -3, -5), and C(3, -4, -4) form the vertices of a right-angled triangle.

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medium

For any two vectors

\vec{a}

and

\vec{b}

, prove the Lagrange's identity:

|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2

.

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medium

Recall the formula for the section formula for internal division. Find the position vector of a point R which divides the line joining two points P and Q, with position vectors

\vec{p}

and

\vec{q}

respectively, in the ratio 3:2 internally.

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medium

Calculate the scalar product of the vectors

\vec{a} = 5\hat{i} - \hat{j} + 2\hat{k}

and

\vec{b} = \hat{i} + 3\hat{j} - \hat{k}

.

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medium

Calculate the vector product

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

for

\vec{a} = 2\hat{i} + \hat{j}

and

\vec{b} = 3\hat{j} - \hat{k}

.

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medium

The diagonals of a parallelogram are given by the vectors

\vec{d_1} = 3\hat{i} + \hat{j} - 2\hat{k}

and

\vec{d_2} = \hat{i} - 3\hat{j} + 4\hat{k}

. Calculate the area of the parallelogram.

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medium

Propose a method using vector products to determine if four points with position vectors

\vec{a}

,

\vec{b}

,

\vec{c}

, and

\vec{d}

are coplanar.

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medium

Justify whether the condition for equality holds in the triangle inequality, i.e.,

|\vec{a} + \vec{b}| = |\vec{a}| + |\vec{b}|

, for two non-zero vectors.

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medium

The position vectors of two points P and Q are

2\hat{i} + \hat{j} - 3\hat{k}

and

3\hat{i} - 2\hat{j} + \hat{k}

respectively. Calculate the position vector of a point R which divides the line segment PQ internally in the ratio 1:3.

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medium

Define 'direction cosines' of a vector. If a vector has direction ratios

a, b, c

and magnitude

r

, state the relationship between the direction cosines (

l, m, n

) and these quantities.

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medium

In a regular hexagon ABCDEF, if

\overrightarrow{AB} = \vec{a}

and

\overrightarrow{BC} = \vec{b}

, derive the vectors for the remaining sides

\overrightarrow{CD}

,

\overrightarrow{DE}

,

\overrightarrow{EF}

, and

\overrightarrow{FA}

in terms of

\vec{a}

and

\vec{b}

.

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medium

Recall the condition for two vectors

\vec{a}

and

\vec{b}

to be collinear in terms of a scalar multiple.

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medium

Describe the concept of a 'position vector' of a point P(x, y, z) in a three-dimensional Cartesian coordinate system. Also, write the formula for its magnitude.

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medium

The diagonals of a parallelogram are given by vectors

\vec{d_1} = 2\hat{i} - 3\hat{j} + 5\hat{k}

and

\vec{d_2} = 4\hat{i} + \hat{j} - \hat{k}

. Formulate the vectors representing the adjacent sides of the parallelogram and then evaluate its area.

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hard

Calculate a vector

\vec{d}

which is perpendicular to both vectors

\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}

and

\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}

, and satisfies the condition

\vec{c} \cdot \vec{d} = 18

, where

\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}

.

Try solving it in your notebook first, then check the solution.

hard

Given the vectors

\vec{\alpha} = \hat{i} + 2\hat{j} - \hat{k}

and

\vec{\beta} = 4\hat{i} + 4\hat{j} + 7\hat{k}

. Express the vector

\vec{\beta}

in the form

\vec{\beta} = \vec{\beta_1} + \vec{\beta_2}

, where

\vec{\beta_1}

is parallel to

\vec{\alpha}

and

\vec{\beta_2}

is perpendicular to

\vec{\alpha}

.

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hard

If

\vec{a}

,

\vec{b}

, and

\vec{c}

are three unit vectors such that

\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0

and the angle between

\vec{b}

and

\vec{c}

is

\frac{\pi}{3}

, justify that

\vec{a} = \pm \frac{2}{\sqrt{3}}(\vec{b} \times \vec{c})

.

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hard

Evaluate the expression

|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2

for any vector

\vec{a}

. Justify your result.

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hard

A student argues that if

\vec{a} \times \vec{b} = \vec{a} \times \vec{c}

and

\vec{a} \neq \vec{0}

, then it must be true that

\vec{b} = \vec{c}

. Critique this argument and formulate the correct conclusion.

Try solving it in your notebook first, then check the solution.

hard

Justify that the shortest distance between two skew lines, with vector equations

\vec{r} = \vec{a_1} + \lambda \vec{b_1}

and

\vec{r} = \vec{a_2} + \mu \vec{b_2}

, is the projection of the vector joining a point on each line onto the vector normal to both lines. Formulate the final expression for this distance.

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hard

Explain the concept of scalar multiplication of a vector. Describe how multiplying a vector

\vec{a}

by a scalar

\lambda

affects its magnitude and direction for the cases when (i)

\lambda > 0

, (ii)

\lambda < 0

, and (iii)

\lambda = 0

.

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hard

Summarize the properties of the vector (cross) product with respect to commutativity and collinearity.

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hard

Calculate a vector of magnitude 10 units that is parallel to the resultant of vectors

\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}

and

\vec{b} = \hat{i} - 3\hat{j} - \hat{k}

.

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hard

If

\vec{a}, \vec{b}, \vec{c}

are three vectors such that

\vec{a}+\vec{b}+\vec{c}=\vec{0}

,

|\vec{a}|=5

,

|\vec{b}|=3

, and

|\vec{c}|=7

, calculate the value of

\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}

.

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hard

Define the scalar (dot) product and the vector (cross) product of two vectors

\vec{a}

and

\vec{b}

. Summarize the key differences between the two products in terms of the nature of the result and the geometric condition under which the result is zero.

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hard

Create a proof to show that the medians of a triangle are concurrent. Let the vertices of the triangle be A, B, and C with position vectors

\vec{a}

,

\vec{b}

, and

\vec{c}

respectively.

Try solving it in your notebook first, then check the solution.

Vector AlgebraClass 12 Mathematics Practice Questions