Vector Algebra

Previous Years' CBSE Board Questions

10.2 Some Basic Concepts

V5A (1 mark)

1. Find a vector <math>\vec{a}</math> of magnitude <math>5\sqrt{2}</math>, making an angle of <math>\frac{\pi}{4}</math> with x-axis, <math>\frac{\pi}{2}</math> with y-axis and an acute angle <math>\theta</math> with z-axis. (AI 2014) EV

SAI (2 marks)

Find a vector <math>\vec{r}</math> equally inclined to the three axes and whose magnitude is <math>3\sqrt{3}</math> units. (2020) An

10.3 Types of Vectors

MCO

(Delhi 2014)

1. Two vectors <math>\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}</math> and <math>\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}</math> are collinear if
2. (c) a₁ = b₁, a₂ = b₂, a₃ = b₃ <math>a_1b_1 + a_2b_2 + a_3b_3 = 0</math> (b) <math>\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}</math>
3. <math>a_1 + a_2 + a_3 = b_1 + b_2 + b_3</math>

1. The value of p for which <math>p(i+j+k)</math> is a unit vector is
2. 0 (b) <math>\frac{1}{\sqrt{2}}</math> (c) 1 (d) √3 (2020) Ap

10.4 Addition of Vectors

MCQ

1. ABCD is a rhombus, whose diagonals intersect at E. Then EA+EB+EC+ED equals
2. Ö (b) AD (c) 2BC (d) 2AD (2020) Ap

10.5 Multiplication of a Vector by a Scalar

MCQ

A unit vector along the vector <math>4\hat{i} - 3\hat{k}</math> is

1. <math>\frac{1}{7}(4\hat{i}-3\hat{k})</math> (b) <math>\frac{1}{5}(4\hat{i}-3\hat{k})</math>
2. <math>\frac{1}{\sqrt{2}}(4\hat{i}-3\hat{k})</math> (d) <math>\frac{1}{\sqrt{\epsilon}}(4\hat{i}-3\hat{k})</math>

VSA (1 mark)

The position vector of two points A and B are <math>\overline{OA} = 2i - j - k</math> and <math>\overline{OB} = 2i - j + 2k</math>, respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2:1 is _

1. Find the position vector of a point which divides the join of points with position vectors <math>\vec{a} = 2\vec{b}</math> and <math>2\vec{a} + \vec{b}</math> externally in the ratio 2:1. (Delhi 2016) An
2. Write the position vector of the point which divides the join of points with position vectors <math>3\vec{a}-2\vec{b}</math> and 2ā+3b in the ratio 2:1. (Al 2016)
3. Find the unit vector in the direction of the sum of the vectors <math>2\hat{i} + 3\hat{j} - \hat{k}</math> and <math>4\hat{i} - 3\hat{j} + 2\hat{k}</math>. (Foreign 2015)
4. Find a vector in the direction of <math>\vec{a} = \hat{i} - 2\hat{j}</math> that has magnitude 7 units. (Delhi 2015C)

Write the direction ratios of the vector 3a+2b where

<math>\vec{a} = \hat{i} + \hat{j} - 2\hat{k}</math> and <math>\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}</math>. (AI 2015C) An

1. Write a unit vector in the direction of the sum of the vectors <math>\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}</math> and <math>\vec{b} = 2\hat{i} + \hat{j} - 7\hat{k}</math>.
2. Find the value of 'p' for which the vectors <math>3\hat{i} + 2\hat{j} + 9\hat{k}</math> and <math>\hat{i} = 2p\hat{j} + 3\hat{k}</math> are parallel. (Al 2014) An

Find a vector in the direction of vector 2i - 3j + 6k

which has magnitude 21 units. (Foreign 2014)

Write a unit vector in the direction of vector PQ.

where P and Q are the points(1, 3, 0) and (4, 5, 6)

respectively. (Foreign 2014)

Write a vector in the direction of the vector i - 2j + 2k

that has magnitude 9 units. (Delhi 2014C) An

SA I (2 marks)

X and Y are two points with position vectors 3a+b

and <math>\vec{a}-3\vec{b}</math> respectively. Write the position vector of a

point Z which divides the line segment XY in the ratio

2:1 externally. (Al 2019) Cr

(4 marks)

The two vectors j+k and 3i-j+4k represent the

two sides AB and AC, respectively of a <math>\triangle</math>ABC. Find the

length of the median through A.

(Delhi 2016, Foreign 2015)

10.6Product of Two Vectors

MCO

20. If <math>\theta</math> is the angle between two vectors <math>\vec{a}</math> and <math>\vec{b}</math>, then

<math>\vec{a} \cdot \vec{b} \ge 0</math> only when <math>0 < \theta < \frac{\pi}{2}</math> <math>0 \le \theta \le \frac{\pi}{}</math>

1. 0 < θ < π (d) 0≤θ≤π (2023)

21. The magnitude of the vector 6î - 2ĵ+3k is

1. 1 (b) 5 (c) 7 (d) 12

(2020) An (2023)

1. If the projection of <math>\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}</math> on <math>\vec{b} = 2\hat{i} + \lambda\hat{k}</math> is zero, then the value of <math>\lambda</math> is
2. U (b) 1
3. <u>-2</u> (d) <math>\frac{-3}{2}</math>

1. If î, ĵ, k are unit vectors along three mutually perpendicular directions, then
2. <math>\hat{i} \cdot \hat{j} = 1</math> (b) <math>\hat{i} \times \hat{j} = 1</math>
3. <math>\hat{i} \cdot \hat{k} = 0</math> (d) <math>\hat{i} \times \hat{k} = 0</math>

VSA (1 mark)

Find the magnitude of vector a given by <math>\vec{a} = (\hat{i} + 3\hat{j} - 2\hat{k}) \times (-\hat{i} + 3\hat{k})</math> (2021C)

1. If <math>\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}</math> and <math>\vec{b} = 5\hat{i} - 3\hat{j} - 4\hat{k}</math> then find the ratio

projection of vector a on vector b (2020C) projection of vector b on vector a

(Foreign 2014)

The area of the parallelogram whose diagonals are 2i and -3k is _____ square units. (2020)

The value of λ for which the vectors 2î - λĵ + k and i + 2j – k are orthogonal is _____. (2020)

Find the magnitude of each of the two vectors \(\vec{a}\) and \(\vec{b}\). having the same magnitude such that the angle between them is 60° and their scalar product is <math>\frac{y}{z}</math>.

(2018) An

1. Write the number of vectors of unit length perpendicular to both the vectors <math>\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}</math> and <math>\vec{b} = j + k</math> (Al 2016)
2. If <math>\vec{a}, \vec{b}, \vec{c}</math> are unit vectors such that <math>\vec{a} + \vec{b} + \vec{c} = \vec{0}</math>, then write the value of <math>\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}</math>.

(AI 2014C) Cr

(NCERT, Foreign 2016)

1. If <math>|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 400</math> and <math>|\vec{a}| = 5</math> then write the value of IbI. (Foreign 2016)
2. If <math>\vec{a} = 7\hat{i} + \hat{j} - 4\hat{k}</math> and <math>\vec{b} = 2\hat{i} + 6\hat{j} + 3\hat{k}</math>, then find the projection of a onb. (Delhi 2015)
3. If â, b and ĉ are mutually perpendicular unit vectors, then find the value of <math>12\hat{a} + \hat{b} + \hat{c}1</math>. (Al 2015) Ev

Write a unit vector perpendicular to both the vectors <math>\vec{a} = \hat{i} + \hat{j} + \hat{k}</math> and <math>\vec{b} = \hat{i} + \hat{j}</math>. (Al 2015)

1. Find the area of a parallelogram whose adjacent sides are represented by the vectors <math>2\hat{i} - 3\hat{k}</math> and <math>4\hat{j} + 2\hat{k}</math>. (Foreign 2015)
2. If <math>\vec{a}</math> and <math>\vec{b}</math> are unit vectors, then what is the angle between <math>\vec{a}</math> and <math>\vec{b}</math> so that <math>\sqrt{2}\vec{a} - \vec{b}</math> is a unit vector?
3. Find the projection of the vector <math>\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}</math> on the vector <math>\vec{b} = 2\vec{i} + 2\vec{j} + \vec{k}</math>. (AI 2015C)
4. Find the projection of vector <math>\hat{i}+3\hat{j}+7\hat{k}</math> on the vector <math>2\hat{i} - 3\hat{j} + 6\hat{k}</math>. (Delhi 2014) (U)

If a and b are two unit vectors such that a+b is also

a unit vector, then find the angle between <math>\vec{a}</math> and <math>\vec{b}</math>

(Delhi 2014) (Ap

1. If vectors <math>\vec{a}</math> and <math>\vec{b}</math> are such that, <math>|\vec{a}|=3, |\vec{b}|=\frac{2}{3}</math> and <math>\vec{a}\times\vec{b}</math> is a unit vector, then write the angle between <math>\vec{a}</math> and <math>\vec{b}</math>. (Delhi 2014) Cr
2. If <math>\vec{a}</math> and <math>\vec{b}</math> are perpendicular vectors, <math>|\vec{a}+\vec{b}|=13</math> and |a|=5, find the value of |b| (Al 2014) An
3. Write the projection of the vector <math>\hat{i} + \hat{j} + \hat{k}</math> along the vector 1. (Foreign 2014)

Write the value of îx(ĵ+k)+ĵx(k+l)+kx(l+ĵ).

1. Write the projection of the vector \(\bar{a} = 2\hat{i} - \hat{j} + \hat{k}\) on the vector <math>\vec{b} = \hat{i} + 2\hat{i} + 2\hat{k}</math>. (Delhi 2014C) (Ap)
2. If <math>\vec{a}</math> and <math>\vec{b}</math> are unit vectors, then find the angle vector. between <math>\vec{a}</math> and <math>\vec{b}</math>, given that <math>(\sqrt{3}\vec{a} - \vec{b})</math> is a unit (Delhi 2014C)
3. Write the value of cosine of the angle which the vector <math>\vec{a} = \vec{i} + \vec{j} + \vec{k}</math> makes with y-axis.

(Delhi 2014C) (Ap)

1. If <math>|\vec{a}|=8</math>, <math>|\vec{b}|=3</math> and <math>|\vec{a}\times\vec{b}|=12</math>, find the angle between <math>\vec{a}</math> and <math>\vec{b}</math>. (AI 2014C)
2. Find the angle between x-axis and the vector <math>\hat{i} + \hat{j} + \hat{k}</math>.

SAI (2 marks)

1. If <math>\vec{a}=4\hat{i}-\hat{j}+\hat{k}</math> and <math>\vec{b}=2\hat{i}-2\hat{j}+\hat{k}</math>, then find a unit vector along the vector <math>\vec{a} \times \vec{b}</math>. (2023)
2. If the vectors <math>\vec{a}</math> and <math>\vec{b}</math> are such that <math>|\vec{a}| = 3</math>, <math>|\vec{b}| = \frac{2}{3}</math> and <math>\vec{a} \times \vec{b}</math> is a unit vector, then find the angle between <math>\vec{a}</math> and <math>\vec{b}</math>. (2023)
3. Find the area of a parallelogram whose adjacent sides are determined by the vectors <math>\vec{a} = \hat{i} - \hat{j} + 3\hat{k}</math> and <math>\vec{b}=2\hat{i}-7\hat{i}+\hat{k}</math>. (2023)
4. Write the projection of the vector <math>(\vec{b}+\vec{c})</math> on the vector <math>\vec{a}</math> where, <math>\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}</math>, <math>\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}</math> and <math display="block">\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}.</math> (Term II, 2021-22) Ap

53. If <math>\vec{a} = \hat{i} + \hat{j} - 2\hat{k}</math> and <math>\vec{b} = -\hat{i} + 2\hat{j} + 2\hat{k}</math> and <math>\vec{c} = -\hat{i} + 2\hat{j} - \hat{k}</math> are

three vectors, then find a vector perpendicular to

both the vectors <math>(\vec{a}+\vec{b})</math> and <math>(\vec{b}-\vec{c})</math>. (Term II, 2021-22C)

(Delhi 2015C) An