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Mathematics · Class 12 Science

Ch 10Vector Algebra — Class 12 Mathematics, concept-first.

Every day, we encounter questions that demand different kinds of answers. "What is your height?" expects a single number — say, 1.6 metres. This number, a real value with only magnitude, is called a scalar. Scalars are quantities that are completely described by a single real number.

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Q&A

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Concepts

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Key concepts

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Vector Displacement

Displacement is the straight-line change in position from a starting point to an ending point — captured as a single vector that carries both how far and in which direction.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

10.1

Introduction

Every day, we encounter questions that demand different kinds of answers. "What is your height?" expects a single number — say, 1.6 metres.

10.2

Some Basic Concepts

Consider any straight line in a plane or in three-dimensional space. This line can be traversed in two opposite directions, indicated by arrowheads.

10.3

Types of Vectors

8 Q

Vectors are classified into several types based on their magnitude, direction, and relative positions.

+Worked Examplesi3 questions
  1. Example 1Represent graphically a displacement of 40 km, $30^\circ$ west of south. ![Displacement of 40 km, 30 degrees west of south](/figures/vector-…Free
  2. Example 2Classify the following measures as scalars and vectors. (i) 5 seconds (ii) 1000 $\text{cm}^3$ (iii) 10 Newton (iv) 30 km/hr (v) 10 $\text{g/…Preview
  3. Example 3In Fig 10.5, which of the vectors are: (i) Collinear (ii) Equal (iii) Coinitial. ![Four vectors a, b, c, d used to identify collinear and co…Preview
+Exercise 10.1i5 questions
  1. Q1Represent graphically a displacement of 40 km, $30^\circ$ east of north.Free
  2. Q2Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 metres north-west (iii) $40^\circ$ (iv) 40 watt (v) $10^{-19}$ coul…Free
  3. Q3Classify the following as scalar and vector quantities. (i) time period (ii) distance (iii) force (iv) velocity (v) work donePreview
  4. Q4In the figure (Fig 10.6), four vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$ are shown. Identify the following vectors: (i) Coinitia…Preview
  5. Q5Answer the following as true or false. (i) $\vec{a}$ and $-\vec{a}$ are collinear. (ii) Two collinear vectors are always equal in magnitude.…Preview
10.4

Addition of Vectors

When a girl moves from A to B, and then from B to C, her net displacement from A to C is the vector sum of the two individual displacements. This leads to the triangle law of vector addition.

10.5

Multiplication of a Vector by a Scalar

Multiplying a vector by a number (a scalar) scales it: the vector's direction either stays the same or flips, and its length changes by a factor equal to the absolute value of that number.

10.5.1

Components of a Vector

To calculate with vectors precisely, we describe any vector by numbers — its components along the coordinate axes. We first define the three unit vectors along the axes.

+Worked Examplesi6 questions
  1. Example 4Find the values of $x$, $y$ and $z$ so that the vectors $\vec{a}=x\hat{i}+2\hat{j}+z\hat{k}$ and $\vec{b}=2\hat{i}+y\hat{j}+\hat{k}$ are equ…Free
  2. Example 5Let $\vec{a}=\hat{i}+2\hat{j}$ and $\vec{b}=2\hat{i}+\hat{j}$. Is $|\vec{a}|=|\vec{b}|$? Are the vectors $\vec{a}$ and $\vec{b}$ equal?Free
  3. Example 6Find unit vector in the direction of vector $\vec{a}=2\hat{i}+3\hat{j}+\hat{k}$.Preview
  4. Example 7Find a vector in the direction of vector $\vec{a}=\hat{i}-2\hat{j}$ that has magnitude 7 units.Preview
  5. Example 8Find the unit vector in the direction of the sum of the vectors, $\vec{a}=2\hat{i}+2\hat{j}-5\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}+3\hat{k}…Preview
  6. Example 9Write the direction ratios of the vector $\vec{a}=\hat{i}+\hat{j}-2\hat{k}$ and hence calculate its direction cosines.Preview
10.5.2

Vector Joining Two Points

In coordinate geometry, every point in space is represented by its position vector relative to the origin.

+Worked Examplesi1 question
  1. Example 10Find the vector joining the points $P(2, 3, 0)$ and $Q(-1, -2, -4)$ directed from P to Q.Preview
10.5.3

Section Formula

21 Q

Two points and have position vectors and . A point on the line joining and has position vector . We find when divides the segment in a given ratio — either internally ( between and ) or externally ( o…

+Worked Examplesi2 questions
  1. Example 11Consider two points P and Q with position vectors $\overrightarrow{OP}=3\vec{a}-2\vec{b}$ and $\overrightarrow{OQ}=\vec{a}+\vec{b}$. Find th…Free
  2. Example 12Show that the points $A(2\hat{i}-\hat{j}+\hat{k})$, $B(\hat{i}-3\hat{j}-5\hat{k})$, $C(3\hat{i}-4\hat{j}-4\hat{k})$ are the vertices of a ri…Preview
+Exercise 10.2i19 questions
  1. Q1Compute the magnitude of the following vectors: $\vec{a} = \hat{i} + \hat{j} + \hat{k}$; $\vec{b} = 2\hat{i} - 7\hat{j} - 3\hat{k}$; $\vec{c…Free
  2. Q2Write two different vectors having same magnitude.Free
  3. Q3Write two different vectors having same direction.Free
  4. Q4Find the values of $x$ and $y$ so that the vectors $2\hat{i} + 3\hat{j}$ and $x\hat{i} + y\hat{j}$ are equal.Preview
  5. Q5Find the scalar and vector components of the vector with initial point $(2, 1)$ and terminal point $(-5, 7)$.Preview
  6. Q6Find the sum of the vectors $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$ and $\vec{c} = \hat{i} - 6…Preview
  7. Q7Find the unit vector in the direction of the vector $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$.Preview
  8. Q8Find the unit vector in the direction of vector $\vec{PQ}$, where P and Q are the points $(1, 2, 3)$ and $(4, 5, 6)$, respectively.Preview
  9. Q9For given vectors, $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = -\hat{i} + \hat{j} - \hat{k}$, find the unit vector in the direc…Preview
  10. Q10Find a vector in the direction of vector $5\hat{i} - \hat{j} + 2\hat{k}$ which has magnitude 8 units.Preview
  11. Q11Show that the vectors $2\hat{i} - 3\hat{j} + 4\hat{k}$ and $-4\hat{i} + 6\hat{j} - 8\hat{k}$ are collinear.Preview
  12. Q12Find the direction cosines of the vector $\hat{i} + 2\hat{j} + 3\hat{k}$.Preview
  13. Q13Find the direction cosines of the vector joining the points A $(1, 2, -3)$ and B $(-1, -2, 1)$, directed from A to B.Preview
  14. Q14Show that the vector $\hat{i} + \hat{j} + \hat{k}$ is equally inclined to the axes OX, OY and OZ.Preview
  15. Q15Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \ha…Preview
  16. Q16Find the position vector of the mid point of the vector joining the points $P(2, 3, 4)$ and $Q(4, 1, -2)$.Preview
  17. Q17Show that the points A, B and C with position vectors, $\vec{a} = 3\hat{i} - 4\hat{j} - 4\hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$…Preview
  18. Q18In triangle $ABC$ (Fig 10.18), which of the following is not true: (A) $\vec{AB} + \vec{BC} + \vec{CA} = \vec{0}$ (B) $\vec{AB} + \vec{BC} -…Preview
  19. Q19If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following are incorrect: (A) $\vec{b} = \lambda\vec{a}$, for some sc…Preview
10.6

Product of Two Vectors

So far we have studied addition and subtraction of vectors. Multiplication of two vectors is defined in two ways — one where the result is a scalar (the scalar or dot product), and another where the r…

10.6.1

Scalar (or Dot) Product of Two Vectors

The scalar product (also called the dot product) produces a real number — a scalar — from two vectors. It measures how much of one vector points in the direction of the other.

10.6.2

Projection of a Vector on a Line

27 Q

When a vector makes an angle with a directed line (measured anticlockwise), the projection vector of on is a vector whose magnitude is , with the same direction as when and opposite when .

+Worked Examplesi9 questions
  1. Example 13Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes 1 and 2 respectively and when $\vec{a}\cdot\vec{b}=1$.Free
  2. Example 14Find angle $\theta$ between the vectors $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$.Free
  3. Example 15If $\vec{a}=5\hat{i}-\hat{j}-3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}-5\hat{k}$, then show that the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\v…Free
  4. Example 16Find the projection of the vector $\vec{a}=2\hat{i}+3\hat{j}+2\hat{k}$ on the vector $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$.Preview
  5. Example 17Find $|\vec{a}-\vec{b}|$, if two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{a}\cdot\vec{b}=4$.Preview
  6. Example 18If $\vec{a}$ is a unit vector and $(\vec{x}-\vec{a})\cdot(\vec{x}+\vec{a})=8$, then find $|\vec{x}|$.Preview
  7. Example 19For any two vectors $\vec{a}$ and $\vec{b}$, we always have $|\vec{a}\cdot\vec{b}|\le|\vec{a}|\,|\vec{b}|$ (Cauchy-Schwartz inequality).Preview
  8. Example 20For any two vectors $\vec{a}$ and $\vec{b}$, we always have $|\vec{a}+\vec{b}|\le|\vec{a}|+|\vec{b}|$ (triangle inequality).Preview
  9. Example 21Show that the points $A(-2\hat{i}+3\hat{j}+5\hat{k})$, $B(\hat{i}+2\hat{j}+3\hat{k})$ and $C(7\hat{i}-\hat{k})$ are collinear.Preview
+Exercise 10.3i18 questions
  1. Q1Find the angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\sqrt{3}$ and $2$, respectively having $\vec{a} \cdot \vec{b}=\s…Free
  2. Q2Find the angle between the vectors $\hat{i}-2\hat{j}+3\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k}$.Free
  3. Q3Find the projection of the vector $\hat{i}-\hat{j}$ on the vector $\hat{i}+\hat{j}$.Free
  4. Q4Find the projection of the vector $\hat{i}+3\hat{j}+7\hat{k}$ on the vector $7\hat{i}-\hat{j}+8\hat{k}$.Preview
  5. Q5Show that each of the given three vectors is a unit vector: $\frac{1}{7}(2\hat{i}+3\hat{j}+6\hat{k})$, $\frac{1}{7}(3\hat{i}-6\hat{j}+2\hat{…Preview
  6. Q6Find $|\vec{a}|$ and $|\vec{b}|,$ if $(\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b})=8$ and $|\vec{a}|=8|\vec{b}|.$Preview
  7. Q7Evaluate the product $(3\vec{a}-5\vec{b}) \cdot (2\vec{a}+7\vec{b}).$Preview
  8. Q8Find the magnitude of two vectors $\vec{a}$ and $\vec{b},$ having the same magnitude and such that the angle between them is $60^\circ$ and…Preview
  9. Q9Find $|\vec{x}|,$ if for a unit vector $\vec{a}, (\vec{x}-\vec{a}) \cdot (\vec{x}+\vec{a})=12.$Preview
  10. Q10If $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}, \vec{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\lambda\vec…Preview
  11. Q11Show that $|\vec{a}|\vec{b}+|\vec{b}|\vec{a}$ is perpendicular to $|\vec{a}|\vec{b}-|\vec{b}|\vec{a},$ for any two nonzero vectors $\vec{a}$…Preview
  12. Q12If $\vec{a} \cdot \vec{a}=0$ and $\vec{a} \cdot \vec{b}=0,$ then what can be concluded about the vector $\vec{b}?$Preview
  13. Q13If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0},$ find the value of $\vec{a} \cdot \vec{b}+\vec{b…Preview
  14. Q14If either vector $\vec{a}=\vec{0}$ or $\vec{b}=\vec{0},$ then $\vec{a} \cdot \vec{b}=0.$ But the converse need not be true. Justify your ans…Preview
  15. Q15If the vertices A, B, C of a triangle ABC are $(1, 2, 3), (-1, 0, 0), (0, 1, 2),$ respectively, then find $\angle ABC. [\angle ABC$ is the a…Preview
  16. Q16Show that the points A$(1, 2, 7),$ B$(2, 6, 3)$ and C$(3, 10, -1)$ are collinear.Preview
  17. Q17Show that the vectors $2\hat{i}-\hat{j}+\hat{k}, \hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$ form the vertices of a right an…Preview
  18. Q18If $\vec{a}$ is a nonzero vector of magnitude '$a$' and $\lambda$ a nonzero scalar, then $\lambda\vec{a}$ is a unit vector if (A) $\lambda =…Preview
10.6.3

Vector (or Cross) Product of Two Vectors

16 Q

In a right-handed rectangular coordinate system, if you curl the fingers of your right hand from the positive x-axis toward the positive y-axis, your thumb points along the positive z-axis.

+Worked Examplesi4 questions
  1. Example 22Find $|\vec{a}\times\vec{b}|$, if $\vec{a}=2\hat{i}+\hat{j}+3\hat{k}$ and $\vec{b}=3\hat{i}+5\hat{j}-2\hat{k}$.Free
  2. Example 23Find a unit vector perpendicular to each of the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, where $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\…Free
  3. Example 24Find the area of a triangle having the points $A(1, 1, 1)$, $B(1, 2, 3)$ and $C(2, 3, 1)$ as its vertices.Preview
  4. Example 25Find the area of a parallelogram whose adjacent sides are given by the vectors $\vec{a}=3\hat{i}+\hat{j}+4\hat{k}$ and $\vec{b}=\hat{i}-\hat…Preview
+Exercise 10.4i12 questions
  1. Q1Find $|\vec{a} \times \vec{b}|$, if $\vec{a}=\hat{i}-7\hat{j}+7\hat{k}$ and $\vec{b}=3\hat{i}-2\hat{j}+2\hat{k}$.Free
  2. Q2Find a unit vector perpendicular to each of the vector $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, where $\vec{a}=3\hat{i}+2\hat{j}+2\hat{k}$ a…Free
  3. Q3If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$, $\frac{\pi}{4}$ with $\hat{j}$ and an acute angle $\theta$ with $\ha…Free
  4. Q4Show that $(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=2(\vec{a}\times\vec{b})$.Preview
  5. Q5Find $\lambda$ and $\mu$ if $(2\hat{i}+6\hat{j}+27\hat{k})\times(\hat{i}+\lambda\hat{j}+\mu\hat{k})=\vec{0}$.Preview
  6. Q6Given that $\vec{a}\cdot\vec{b}=0$ and $\vec{a}\times\vec{b}=\vec{0}$. What can you conclude about the vectors $\vec{a}$ and $\vec{b}$?Preview
  7. Q7Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be given as $a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, $b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$, $c_1\hat{i}+…Preview
  8. Q8If either $\vec{a}=\vec{0}$ or $\vec{b}=\vec{0}$, then $\vec{a}\times\vec{b}=\vec{0}$. Is the converse true? Justify your answer with an exa…Preview
  9. Q9Find the area of the triangle with vertices $A(1, 1, 2)$, $B(2, 3, 5)$ and $C(1, 5, 5)$.Preview
  10. Q10Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a}=\hat i-\hat j+3\hat k$ and $\vec{b}=2\hat i-7…Preview
  11. Q11Let the vectors $\vec{a}$ and $\vec{b}$ be such that $|\vec{a}|=3$ and $|\vec{b}|=\dfrac{\sqrt{2}}{3}$, then $\vec{a}\times\vec{b}$ is a uni…Preview
  12. Q12Area of a rectangle having vertices A, B, C and D with position vectors $-\hat i+\dfrac{1}{2}\hat j+4\hat k$, $\hat i+\dfrac{1}{2}\hat j+4\h…Preview

Miscellaneous Examples

+Miscellaneous Examplesi5 questions
  1. Example 26Write all the unit vectors in XY-plane.Free
  2. Example 27If $\hat{i}+\hat{j}+\hat{k}$, $2\hat{i}+5\hat{j}$, $3\hat{i}+2\hat{j}-3\hat{k}$ and $\hat{i}-6\hat{j}-\hat{k}$ are the position vectors of p…Free
  3. Example 28Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $|\vec{a}|=3$, $|\vec{b}|=4$, $|\vec{c}|=5$ and each one of them being perpen…Preview
  4. Example 29Three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ satisfy the condition $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. Evaluate the quantity $\mu=\vec{a}…Preview
  5. Example 30If with reference to the right handed system of mutually perpendicular unit vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$, $\vec{\alpha}=3\hat{…Preview

Miscellaneous Exercise on Chapter 10

+Miscellaneous Exercisei19 questions
  1. Q1Write down a unit vector in XY-plane, making an angle of $30^\circ$ with the positive direction of x-axis.Free
  2. Q2Find the scalar components and magnitude of the vector joining the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$.Free
  3. Q3A girl walks 4 km towards west, then she walks 3 km in a direction $30^\circ$ east of north and stops. Determine the girl's displacement fro…Free
  4. Q4If $\vec{a} = \vec{b} + \vec{c}$, then is it true that $|\vec{a}| = |\vec{b}| + |\vec{c}|$? Justify your answer.Preview
  5. Q5Find the value of $x$ for which $x(\hat{i} + \hat{j} + \hat{k})$ is a unit vector.Preview
  6. Q6Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \h…Preview
  7. Q7If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} + \hat{k}$, find a u…Preview
  8. Q8Show that the 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.Preview
  9. Q9Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $(2\vec{a} + \vec{b})$ an…Preview
  10. Q10The two adjacent sides of a parallelogram are $2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\hat{i} - 2\hat{j} - 3\hat{k}$. Find the unit vector par…Preview
  11. Q11Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\pm \left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}…Preview
  12. Q12Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$. Fin…Preview
  13. Q13The scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with a unit vector along the sum of vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$…Preview
  14. Q14If $\vec{a}$, $\vec{b}$, $\vec{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{a} + \vec{b} + \vec{c}…Preview
  15. Q15Prove that $(\vec{a}+\vec{b})\cdot(\vec{a}+\vec{b})=|\vec{a}|^2+|\vec{b}|^2$, if and only if $\vec{a},\vec{b}$ are perpendicular, given $\ve…Preview
  16. Q16If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a}\cdot\vec{b}\ge 0$ only when (A) $0<\theta<\frac{\pi}{2}$…Preview
  17. Q17Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a}+\vec{b}$ is a unit vector if (A) $\the…Preview
  18. Q18The value of $\hat{i}\cdot(\hat{j}\times\hat{k})+\hat{j}\cdot(\hat{i}\times\hat{k})+\hat{k}\cdot(\hat{i}\times\hat{j})$ is (A) 0 (B) -1 (C)…Preview
  19. Q19If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b}$, then $|\vec{a}\cdot\vec{b}|=|\vec{a}\times\vec{b}|$ when $\theta$…Preview

Summary

- Vector definition: A quantity with both magnitude and direction, denoted as or . Its magnitude is . - Position vector: of point is . - Direction cosines: Cosines of angles with axes: , , , with .

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 43 questions43 questions
  1. Q1Find the unit vector in the direction of sum of vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2\hat{j}+\hat{k}$.Free
  2. Q2If $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-2\hat{k}$, find the unit vector in the direction of (i) $6\vec{b}$ (ii)…Free
  3. Q3Find a unit vector in the direction of $\vec{PQ}$, where P and Q have co-ordinates $(5, 0, 8)$ and $(3, 3, 2)$, respectively.Free
  4. Q4If $\vec{a}$ and $\vec{b}$ are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that…Preview
  5. Q5Using vectors, find the value of $k$ such that the points $(k, -10, 3)$, $(1, -1, 3)$ and $(3, 5, 3)$ are collinear.Preview
  6. Q6A vector $\vec{r}$ is inclined at equal angles to the three axes. If the magnitude of $\vec{r}$ is $2\sqrt{3}$ units, find $\vec{r}$.Preview
  7. Q7A vector $\vec{r}$ has magnitude 14 and direction ratios 2, 3, -6. Find the direction cosines and components of $\vec{r}$, given that $\vec{…Preview
  8. Q8Find a vector of magnitude 6, which is perpendicular to both the vectors $2\hat{i}-\hat{j}+2\hat{k}$ and $4\hat{i}-\hat{j}+3\hat{k}$.Preview
  9. Q9Find the angle between the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $3\hat{i}+4\hat{j}-\hat{k}$.Preview
  10. Q10If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, show that $\vec{a}\times\vec{b}=\vec{b}\times\vec{c}=\vec{c}\times\vec{a}$. Interpret the result geome…Preview
  11. Q11Find the sine of the angle between the vectors $\vec{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}$.Preview
  12. Q12If A, B, C, D are the points with position vectors $\hat{i}+\hat{j}-\hat{k}$, $2\hat{i}-\hat{j}+3\hat{k}$, $2\hat{i}-3\hat{k}$, $3\hat{i}-2\…Preview
  13. Q13Using vectors, find the area of the triangle ABC with vertices $A(1, 2, 3)$, $B(2, -1, 4)$ and $C(4, 5, -1)$.Preview
  14. Q14Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.Preview
  15. Q15Prove that in any triangle ABC, $\cos A=\dfrac{b^2+c^2-a^2}{2bc}$, where $a$, $b$, $c$ are the magnitudes of the sides opposite to the verti…Preview
  16. Q16If $\vec{a}$, $\vec{b}$, $\vec{c}$ determine the vertices of a triangle, show that $\dfrac{1}{2}\left[\vec{b}\times\vec{c}+\vec{c}\times\vec…Preview
  17. Q17Show that area of the parallelogram whose diagonals are given by $\vec{a}$ and $\vec{b}$ is $\dfrac{|\vec{a}\times\vec{b}|}{2}$. Also find t…Preview
  18. Q18If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$, find a vector $\vec{c}$ such that $\vec{a}\times\vec{c}=\vec{b}$ and $\v…Preview
  19. Q19The vector $\vec{a}+\vec{b}$ bisects the angle between the non-collinear vectors $\vec{a}$ and $\vec{b}$ if ________Preview
  20. Q20The vectors $\vec{a}=3\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{b}=-\hat{i}-2\hat{k}$ are the adjacent sides of a parallelogram. The acute angle…Preview
  21. Q21The values of $k$ for which $|k\vec{a}|<|\vec{a}|$ and $k\vec{a}+\dfrac{1}{2}\vec{a}$ is parallel to $\vec{a}$ holds true are ________.Preview
  22. Q22The value of the expression $|\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2$ is ________.Preview
  23. Q23If $|\vec{a}\times\vec{b}|^2+|\vec{a}\cdot\vec{b}|^2=144$ and $|\vec{a}|=4$, then $|\vec{b}|$ is equal to ________.Preview
  24. Q24If $\vec{a}$ is any non-zero vector, then $(\vec{a}\cdot\hat{i})\hat{i}+(\vec{a}\cdot\hat{j})\hat{j}+(\vec{a}\cdot\hat{k})\hat{k}$ equals __…Preview
  25. Q25State True or False: If $|\vec{a}|=|\vec{b}|$, then necessarily it implies $\vec{a}=\pm\vec{b}$.Preview
  26. Q26State True or False: Position vector of a point P is a vector whose initial point is origin.Preview
  27. Q27State True or False: If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, then the vectors $\vec{a}$ and $\vec{b}$ are orthogonal.Preview
  28. Q28State True or False: The formula $|\vec{a}+\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2+2\vec{a}\times\vec{b}$ is valid for non-zero vectors $\vec{a}$…Preview
  29. Q29State True or False: If $\vec{a}$ and $\vec{b}$ are adjacent sides of a rhombus, then $\vec{a}\cdot\vec{b}=0$.Preview
  30. Q30The vector in the direction of the vector $\hat{i}-2\hat{j}+2\hat{k}$ that has magnitude 9 is (A) $\hat{i}-2\hat{j}+2\hat{k}$ (B) $\dfrac{\h…Preview
  31. Q31The position vector of the point which divides the join of points $2\vec{a}-3\vec{b}$ and $\vec{a}+\vec{b}$ in the ratio 3 : 1 is (A) $\dfra…Preview
  32. Q32The vector having initial and terminal points as $(2, 5, 0)$ and $(-3, 7, 4)$, respectively is (A) $-\hat{i}+12\hat{j}+4\hat{k}$ (B) $5\hat{…Preview
  33. Q33The angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\sqrt{3}$ and 4, respectively, and $\vec{a}\cdot\vec{b}=2\sqrt{3}$ is…Preview
  34. Q34Find the value of $\lambda$ such that the vectors $\vec{a}=2\hat{i}+\lambda\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+3\hat{k}$ are orth…Preview
  35. Q35The value of $\lambda$ for which the vectors $3\hat{i}-6\hat{j}+\hat{k}$ and $2\hat{i}-4\hat{j}+\lambda\hat{k}$ are parallel is (A) $\dfrac{…Preview
  36. Q36The vectors from origin to the points A and B are $\vec{a}=2\hat{i}-3\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}+3\hat{j}+\hat{k}$, respectively…Preview
  37. Q37For any vector $\vec{a}$, the value of $(\vec{a}\times\hat{i})^2+(\vec{a}\times\hat{j})^2+(\vec{a}\times\hat{k})^2$ is equal to (A) $\vec{a}…Preview
  38. Q38If $|\vec{a}|=10$, $|\vec{b}|=2$ and $\vec{a}\cdot\vec{b}=12$, then value of $|\vec{a}\times\vec{b}|$ is (A) $5$ (B) $10$ (C) $14$ (D) $16$Preview
  39. Q39If $\vec{a}$, $\vec{b}$, $\vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, then the value of $\vec{a}\cdot\vec{b}+\vec…Preview
  40. Q40Projection vector of $\vec{a}$ on $\vec{b}$ is (A) $\left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}$ (B) $\dfrac{\vec{a}\cdot\v…Preview
  41. Q41If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ and $|\vec{a}|=2$, $|\vec{b}|=3$, $|\vec{c}…Preview
  42. Q42If $|\vec{a}|=4$ and $-3\le\lambda\le2$, then the range of $|\lambda\vec{a}|$ is (A) $[0, 8]$ (B) $[-12, 8]$ (C) $[0, 12]$ (D) $[8, 12]$Preview
  43. Q43The number of vectors of unit length perpendicular to the vectors $\vec{a}=2\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$ is (A) o…Preview

CBSE Sample Papers

Questions from official CBSE sample papers.

+Show 7 questions7 questions
  1. Q1If 𝑎⃗ = 3𝑖̂ + 2𝑗̂ + 4𝑘̂ , 𝑏⃗⃗ = 𝑖̂ + 𝑗̂ − 3𝑘̂ and 𝑐⃗ = 6𝑖̂ − 𝑗̂ + 2𝑘̂ are three given vectors, then (2𝑎⃗. 𝑖̂)𝑖̂ − (𝑏⃗⃗. 𝑗̂)𝑗̂ + (𝑐⃗. 𝑘̂)𝑘̂ i…Preview
  2. Q2If |𝑎⃗| = 3, |𝑏⃗⃗| = 4 and |𝑎⃗ + 𝑏⃗⃗| =5, then |𝑎⃗ − 𝑏⃗⃗| = (A) 3 (B) 4 (C) 5 (D) 8Preview
  3. Q3(a) If vectors 𝑎⃗ = 2ı̂ + 2ȷ̂ + 3k̂ , 𝑏⃗⃗ = − ı̂ + 2ȷ̂ + k̂ and 𝑐⃗ = 3ı̂ + ȷ̂ are such that 𝑏⃗⃗ + λ𝑐⃗ is perpendicular to 𝑎⃗ , then find the…Preview
  4. Q4(a) An ant is moving along the vector $\vec{l_1} = \hat{i} - 2\hat{j} + 3\hat{k}$. Few sugar crystals are kept along the vector $\vec{l_2} =…Preview
  5. Q5Find the magnitude of each of two vectors $\vec{a}$ and $\vec{b}$, having the same magnitude such that the angle between them is $60^{\circ}…Preview
  6. Q6If $\theta$ is the angle between two vectors $\hat{i} - 2\hat{j} + \hat{k}$ and $3\hat{i} - 2\hat{j} + \hat{k}$, find $\sin\theta$.Preview
  7. Q7Let $\vec{a} = 4\hat{i} + 5\hat{j} - \hat{k}$, $\vec{b} = \hat{i} - 4\hat{j} + 5\hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j} - \hat{k}$. Find…Preview