JEE Mathematics Vectors and Their Applications MCQs Set A

Vectors and their Applications

Type – 1

Choose the most appropriate option (a, b, c or d).

Question. ABCDEF is a regular hexagon where centre O is the origin. If the position vectors of A and B are \( \vec{i} - \vec{j} + 2\vec{k} \) and \( 2\vec{i} + \vec{j} + \vec{k} \) respectively then \( \vec{BC} \) is equal to
(a) \( \vec{i} - \vec{j} + 2\vec{k} \)
(b) \( -\vec{i} + \vec{j} - 2\vec{k} \)
(c) \( 3\vec{i} + 3\vec{j} - 4\vec{k} \)
(d) None of the options
Answer: (b) \( -\vec{i} + \vec{j} - 2\vec{k} \)

Question. The position vectors of two vertices and the centroid of a triangle are \( \vec{i} + \vec{j}, 2\vec{i} - \vec{j} + 4\vec{k} \) and \( \vec{k} \) respectively. The position vector of the third vertex of the triangle is
(a) \( -3\vec{i} + 2\vec{k} \)
(b) \( 3\vec{i} - 2\vec{k} \)
(c) \( \vec{i} + \frac{2}{3}\vec{k} \)
(d) None of the options
Answer: (a) \( -3\vec{i} + 2\vec{k} \)

Question. Let the position vectors of the points A, B, C be \( \vec{i} + 2\vec{j} + 3\vec{k}, -\vec{i} - \vec{j} + 8\vec{k} \) and \( -4\vec{i} + 4\vec{j} + 6\vec{k} \) respectively. Then the ABC is
(a) right angled
(b) equilateral
(c) isosceles
(d) None of the options
Answer: (b) equilateral

Question. \( \vec{a}, \vec{b}, \vec{c} \) are three vectors of which every pair is noncollinear. If the vector \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) are collinear with \( \vec{c} \) and \( \vec{a} \) respectively then \( \vec{a} + \vec{b} + \vec{c} \) is
(a) a unit vector
(b) the null vector
(c) equally inclined to \( \vec{a}, \vec{b}, \vec{c} \)
(d) None of the options
Answer: (b) the null vector

Question. If \( \vec{r} = 3\vec{i} + 2\vec{j} - 5\vec{k}, \vec{a} = 2\vec{i} - \vec{j} + \vec{k}, \vec{b} = \vec{i} + 3\vec{j} - 2\vec{k} \) and \( \vec{c} = 2\vec{i} + \vec{j} - 3\vec{k} \) such that \( \vec{r} = \lambda\vec{a} + \mu\vec{b} + v\vec{c} \) then
(a) \( \lambda, \mu, v \) are in AP
(b) \( \lambda, \mu, v \) are in AP
(c) \( \lambda, \mu, v \) are in HP
(d) \( \mu, \lambda, v \) are in GP
Answer: (a) \( \lambda, \mu, v \) are in AP

Question. The position vectors of three points are \( 2\vec{a} - \vec{b} + 3\vec{c}, \vec{a} - 2\vec{b} + \lambda\vec{c} \) and \( \mu\vec{a} - 5\vec{b} \) where \( \vec{a}, \vec{b}, \vec{c} \) are noncoplanar vectors. They are collinear when
(a) \( \lambda = -2, \mu = \frac{9}{4} \)
(b) \( \lambda = -\frac{9}{4}, \mu = 2 \)
(c) \( \lambda = \frac{9}{4}, \mu = -2 \)
(d) None of the options
Answer: (c) \( \lambda = \frac{9}{4}, \mu = -2 \)

Question. If \( \vec{a} = \vec{i} + \vec{j} + \vec{k}, \vec{b} = 4\vec{i} + 3\vec{j} + 4\vec{k} \) and \( \vec{c} = \vec{i} + \alpha\vec{j} + \beta\vec{k} \) are linearly dependent vectors and \( |\vec{c}| = \sqrt{3} \) then
(a) \( \alpha = 1, \beta = -1 \)
(b) \( \alpha = 1, \beta = \pm 1 \)
(c) \( \alpha = -1, \beta = \pm 1 \)
(d) \( \alpha = \pm 1, \beta = 1 \)
Answer: (d) \( \alpha = \pm 1, \beta = 1 \)

Question. Let \( \vec{OA} = \vec{a} \) and \( \vec{OB} = \vec{b} \). A vector along one of the bisectors of the angle \( \angle AOB \) is
(a) \( \vec{a} + \vec{b} \)
(b) \( \vec{a} - \vec{b} \)
(c) \( \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} \)
(d) None of the options
Answer: (c) \( \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} \)

Question. A vector has components \( 2p \) and \( 1 \) with respect to a rectangular Cartesian system. The axes are rotated through an angle \( \alpha \) about the origin in the anticlockwise sense. If the vector has components \( p + 1 \) and \( 1 \) with respect to the new system then
(a) \( p = 1, -\frac{1}{3} \)
(b) \( p = 0 \)
(c) \( p = -1, \frac{1}{3} \)
(d) \( p = 1, -1 \)
Answer: (a) \( p = 1, -\frac{1}{3} \)

Question. If \( \vec{a} \) and \( \vec{b} \) are two vectors of magnitude inclined at an angle \( 60^\circ \) then the angle between \( \vec{a} \) and \( \vec{a} + \vec{b} \) is
(a) \( 30^\circ \)
(b) \( 60^\circ \)
(c) \( 45^\circ \)
(d) None of the options
Answer: (a) \( 30^\circ \)

Question. Let \( |\vec{a}| = |\vec{b}| = |\vec{a} - \vec{b}| = 1 \). Then the angle between \( \vec{a} \) and \( \vec{b} \) is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{2} \)
Answer: (b) \( \frac{\pi}{3} \)

Question. A vector of magnitude 4 which is equally inclined to the vectors \( \vec{i} + \vec{j}, \vec{j} + \vec{k} \) and \( \vec{k} + \vec{i} \) is
(a) \( \frac{4}{\sqrt{3}}(\vec{i} - \vec{j} - \vec{k}) \)
(b) \( \frac{4}{\sqrt{3}}(\vec{i} + \vec{j} - \vec{k}) \)
(c) \( \frac{4}{\sqrt{3}}(\vec{i} + \vec{j} + \vec{k}) \)
(d) None of the options
Answer: (c) \( \frac{4}{\sqrt{3}}(\vec{i} + \vec{j} + \vec{k}) \)

Question. If \( \vec{a} + \vec{b} = 2\vec{i} \) and \( 2\vec{a} - \vec{b} = \vec{i} - \vec{j} \) then cosine of the angle between \( \vec{a} \) and \( \vec{b} \) is
(a) \( \sin^{-1} \frac{4}{5} \)
(b) \( \cos^{-1} \frac{4}{5} \)
(c) \( \cos^{-1} \frac{3}{5} \)
(d) None of the options
Answer: (b) \( \cos^{-1} \frac{4}{5} \)

Question. Let \( |\vec{a}| = 1, |\vec{b}| = \sqrt{2}, |\vec{c}| = \sqrt{3} \), and \( \vec{a} \perp (\vec{b} + \vec{c}), \vec{b} \perp (\vec{c} + \vec{a}) \) and \( \vec{c} \perp (\vec{a} + \vec{b}) \). Then \( |\vec{a} + \vec{b} + \vec{c}| \) is
(a) \( \sqrt{6} \)
(b) \( 6 \)
(c) \( \sqrt{14} \)
(d) None of the options
Answer: (a) \( \sqrt{6} \)

Question. \( (\vec{a} \cdot \vec{i})\vec{i} + (\vec{a} \cdot \vec{j})\vec{j} + (\vec{a} \cdot \vec{k})\vec{k} \) is equal to
(a) \( \vec{i} + \vec{j} + \vec{k} \)
(b) \( \vec{a} \)
(c) \( 3\vec{a} \)
(d) None of the options
Answer: (b) \( \vec{a} \)

Question. If \( \vec{a}, \vec{b} \) are unit vectors such that \( \vec{a} + \vec{b} \) is also a unit vector then the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{2\pi}{3} \)
Answer: (d) \( \frac{2\pi}{3} \)

Question. If \( \vec{a} \cdot \vec{i} = \vec{a} \cdot (\vec{i} + \vec{j}) = \vec{a} \cdot (\vec{i} + \vec{j} + \vec{k}) = 1 \) then \( \vec{a} \) is
(a) \( \vec{i} - \vec{j} \)
(b) \( \vec{i} \)
(c) \( \vec{j} \)
(d) \( \vec{k} \)
Answer: (b) \( \vec{i} \)

Question. \( (\vec{a} \cdot \vec{i})^2 + (\vec{a} \cdot \vec{j})^2 + (\vec{a} \cdot \vec{k})^2 \) is equal to
(a) \( \vec{a} \)
(b) \( 3 \)
(c) \( |\vec{a} \cdot (\vec{i} + \vec{j} + \vec{k})|^2 \)
(d) \( \vec{k} \)
Answer: (a) \( \vec{a} \)

Question. \( |\vec{a} + \vec{b}|^2 - |\vec{a} - \vec{b}|^2 \) is equal to
(a) \( 4\vec{a} \cdot \vec{b} \)
(b) \( 0 \)
(c) \( 4|\vec{a} \cdot \vec{b}| \)
(d) None of the options
Answer: (a) \( 4\vec{a} \cdot \vec{b} \)

Question. If \( a, b, c \) are the pth, qth, rth terms of an HP and \( \vec{\mu} = (q - r)\vec{i} + (r - p)\vec{j} + (p - q)\vec{k}, \vec{v} = \frac{\vec{i}}{a} + \frac{\vec{j}}{b} + \frac{\vec{k}}{c} \) then
(a) \( \vec{\mu}, \vec{v} \) are parallel vectors
(b) \( \vec{\mu}, \vec{v} \) are orthogonal vectors
(c) \( \vec{\mu} \cdot \vec{v} = 1 \)
(d) \( \vec{\mu} \times \vec{v} = \vec{i} + \vec{j} + \vec{k} \)
Answer: (b) \( \vec{\mu}, \vec{v} \) are orthogonal vectors

Question. If \( \vec{a} + \vec{b} \perp \vec{a} \) and \( |\vec{b}| = \sqrt{2}|\vec{a}| \) then
(a) \( (2\vec{a} + \vec{b}) \parallel \vec{b} \)
(b) \( (2\vec{a} + \vec{b}) \perp \vec{b} \)
(c) \( (2\vec{a} - \vec{b}) \perp \vec{b} \)
(d) \( (2\vec{a} + \vec{b}) \perp \vec{a} \)
Answer: (b) \( (2\vec{a} + \vec{b}) \perp \vec{b} \)

Question. Let \( \vec{a} = 2\vec{i} + \vec{j} + \vec{k}, \vec{b} = \vec{i} + 2\vec{j} - \vec{k} \) and a unit vector \( \vec{c} \) be coplanar. If \( \vec{c} \) is perpendicular to \( \vec{a} \) then \( \vec{c} = \)
(a) \( \frac{1}{\sqrt{2}}(-\vec{j} + \vec{k}) \)
(b) \( \frac{1}{\sqrt{3}}(-\vec{i} - \vec{j} + \vec{k}) \)
(c) \( \frac{1}{\sqrt{5}}(\vec{i} - 2\vec{j}) \)
(d) \( \frac{1}{\sqrt{3}}(\vec{i} - \vec{j} - \vec{k}) \)
Answer: (a) \( \frac{1}{\sqrt{2}}(-\vec{j} + \vec{k}) \)

Question. Let \( \vec{\lambda} = \vec{a} \times (\vec{b} + \vec{c}), \vec{\mu} = \vec{b} \times (\vec{c} + \vec{a}) \) and \( \vec{v} = \vec{c} \times (\vec{a} + \vec{b}) \). Then
(a) \( \vec{\lambda} + \vec{\mu} = \vec{v} \)
(b) \( \vec{\lambda}, \vec{\mu}, \vec{v} \) are coplanar
(c) \( \vec{\lambda} + \vec{v} = 2\vec{\mu} \)
(d) None of the options
Answer: (b) \( \vec{\lambda}, \vec{\mu}, \vec{v} \) are coplanar

Question. Let \( \vec{a}, \vec{b}, \vec{c} \) be three unit vectors such that \( |\vec{a} + \vec{b} + \vec{c}| = 1, \vec{a} \perp \vec{b} \). If \( \vec{c} \) makes angles \( \alpha, \beta \) with \( \vec{a}, \vec{b} \) respectively then \( \cos \alpha + \cos \beta \) is equal to
(a) \( \frac{3}{2} \)
(b) \( 1 \)
(c) \( -1 \)
(d) None of the options
Answer: (c) \( -1 \)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are three vectors of equal magnitude and the angle between each pair of vectors is \( \frac{\pi}{3} \) such that \( |\vec{a} + \vec{b} + \vec{c}| = \sqrt{6} \) then \( |\vec{a}| \) is equal to
(a) \( 2 \)
(b) \( -1 \)
(c) \( 1 \)
(d) \( \frac{1}{3}\sqrt{6} \)
Answer: (c) \( 1 \)

Question. If \( |\vec{a}| = 5, |\vec{a} - \vec{b}| = 8 \) and \( |\vec{a} + \vec{b}| = 10 \) then \( |\vec{b}| \) is
(a) \( 1 \)
(b) \( \sqrt{57} \)
(c) \( 3 \)
(d) None of the options
Answer: (b) \( \sqrt{57} \)

Question. If \( \vec{a} \) and \( \vec{b} \) are unit vectors and \( \alpha \) is the angle between them then \( \cos \frac{\alpha}{2} \) is
(a) \( \frac{1}{2}|\vec{a} + \vec{b}| \)
(b) \( \frac{1}{2}|\vec{a} - \vec{b}| \)
(c) \( |\vec{a} + \vec{b}| \)
(d) None of the options
Answer: (a) \( \frac{1}{2}|\vec{a} + \vec{b}| \)

Question. If \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \) then
(a) \( \vec{a} \parallel \vec{b} \)
(b) \( \vec{a} \perp \vec{b} \)
(c) \( |\vec{a}| = |\vec{b}| \)
(d) None of the options
Answer: (b) \( \vec{a} \perp \vec{b} \)

Question. Two vectors \( \vec{a} = \vec{i} + \frac{\vec{i}}{\sqrt{3}} \) and \( \vec{b} = \vec{j} + \frac{\vec{i}}{\sqrt{3}} \) are
(a) perpendicular to each other
(b) parallel to each other
(c) inclined to each other at an angle \( \frac{\pi}{3} \)
(d) inclined to each other at an angle \( \frac{\pi}{6} \)
Answer: (d) inclined to each other at an angle \( \frac{\pi}{6} \)

Question. Let \( \vec{a} = 2\vec{i} - \vec{j} + \vec{k}, \vec{b} = \vec{i} + 3\vec{j} - \vec{k} \) and \( \vec{c} = \vec{i} + \vec{j} - 2\vec{k} \). A vector in the plane of \( \vec{b} \) and \( \vec{c} \) whose projection on \( \vec{a} \) has the magnitude \( a \) is
(a) \( 2\vec{i} + 3\vec{j} - 3\vec{k} \)
(b) \( 2\vec{i} + 3\vec{j} + 3\vec{k} \)
(c) \( -2\vec{i} - \vec{j} + 5\vec{k} \)
(d) \( 2\vec{i} + \vec{j} + 5\vec{k} \)
Answer: (c) \( -2\vec{i} - \vec{j} + 5\vec{k} \)

Question. ABC is an equilateral triangle of side a. The value of \( \vec{AB} \cdot \vec{BC} + \vec{BC} \cdot \vec{CA} + \vec{CA} \cdot \vec{AB} \) is equal to
(a) \( \frac{3a^2}{2} \)
(b) \( 3a^2 \)
(c) \( -\frac{3a^2}{2} \)
(d) None of the options
Answer: (c) \( -\frac{3a^2}{2} \)

Question. If \( \vec{a} = \vec{i} + \vec{j}, \vec{b} = 2\vec{j} - \vec{k} \) and \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a}, \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \) then \( \frac{\vec{r}}{|\vec{r}|} \) is equal to
(a) \( \frac{1}{\sqrt{11}}(\vec{i} + 3\vec{j} - \vec{k}) \)
(b) \( \frac{1}{\sqrt{11}}(\vec{i} - 3\vec{j} + \vec{k}) \)
(c) \( \frac{1}{\sqrt{11}}(\vec{i} - \vec{j} + \vec{k}) \)
(d) None of the options
Answer: (a) \( \frac{1}{\sqrt{11}}(\vec{i} + 3\vec{j} - \vec{k}) \)

Question. \( (\vec{a} \times \vec{b})^2 + (\vec{a} \cdot \vec{b})^2 \) is equal to
(a) \( 0 \)
(b) \( |\vec{a}|^2|\vec{b}|^2 \)
(c) \( (|\vec{a}| + |\vec{b}|)^2 \)
(d) \( 1 \)
Answer: (b) \( |\vec{a}|^2|\vec{b}|^2 \)

Question. If \( \vec{p}, \vec{q} \) are two noncollinear and nonzero vector such that \( (b - c)\vec{p} \times \vec{q} + (c - a)\vec{p} + (a - b)\vec{q} = 0 \), where a, b, c are the lengths of the sides of a triangle, then the triangle is
(a) right angled
(b) obtuse angled
(c) equilateral
(d) isosceles
Answer: (c) equilateral

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are any three vectors such that \( (\vec{a} + \vec{b}) \cdot \vec{c} = (\vec{a} - \vec{b}) \cdot \vec{c} = 0 \) then \( (\vec{a} \times \vec{b}) \times \vec{c} \) is
(a) \( \vec{0} \)
(b) \( \vec{a} \)
(c) \( \vec{b} \)
(d) None of the options
Answer: (a) \( \vec{0} \)