JEE Mathematics Vectors and Their Applications MCQs Set 03

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are three noncoplanar nonzero vectors and \( \vec{r} \) is any vector in space then \( (\vec{a} \times \vec{b}) \times (\vec{r} \times \vec{c}) + (\vec{b} \times \vec{c}) \times (\vec{r} \times \vec{a}) + (\vec{c} \times \vec{a}) \times (\vec{r} \times \vec{b}) \) is equal to
(a) \( 2[\vec{a} \vec{b} \vec{c}]\vec{r} \)
(b) \( 3[\vec{a} \vec{b} \vec{c}]\vec{r} \)
(c) \( [\vec{a} \vec{b} \vec{c}]\vec{r} \)
(d) None of the options
Answer: (a) \( 2[\vec{a} \vec{b} \vec{c}]\vec{r} \)

Question. Let \( \vec{a}, \vec{b}, \vec{c} \) be three unit vectors of which \( \vec{b} \) and \( \vec{c} \) are nonparallel. Let the angle between \( \vec{a} \) and \( \vec{b} \) be \( \alpha \) and that between \( \vec{a} \) and \( \vec{c} \) be \( \beta \). If \( \vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2}\vec{b} \) then
(a) \( \alpha = \frac{\pi}{3}, \beta = \frac{\pi}{2} \)
(b) \( \alpha = \frac{\pi}{2}, \beta = \frac{\pi}{3} \)
(c) \( \alpha = \frac{\pi}{6}, \beta = \frac{\pi}{3} \)
(d) None of the options
Answer: (b) \( \alpha = \frac{\pi}{2}, \beta = \frac{\pi}{3} \)

Question. Let \( \vec{a} = 2\vec{i} + \vec{j} - 2\vec{k} \) and \( \vec{b} = \vec{i} + \vec{j} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - \vec{a}| = 2\sqrt{2} \) and the angle between \( \vec{a} \times \vec{b} \) and \( \vec{c} \) is \( 30^\circ \) then \( |(\vec{a} \times \vec{b}) \times \vec{c}| \) is equal to
(a) \( \frac{2}{3} \)
(b) \( \frac{3}{2} \)
(c) 2
(d) 3
Answer: (b) \( \frac{3}{2} \)

Question. Let \( \vec{a} \), and \( \vec{b} \) be two noncollinear unit vectors. If \( \vec{u} = \vec{a} - (\vec{a} \cdot \vec{b})\vec{b} \) and \( \vec{v} = \vec{a} \times \vec{b} \) then \( |\vec{v}| \) is
(a) \( |\vec{u}| \)
(b) \( |\vec{u}| + |\vec{u} \cdot \vec{a}| \)
(c) \( |\vec{u}| + |\vec{u} \cdot \vec{b}| \)
(d) \( |\vec{u}| + \vec{u} \cdot (\vec{a} + \vec{b}) \)
Answer: (a) \( |\vec{u}| \)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \( [\vec{a} \vec{b} \vec{c}] = 4 \) then \( [\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}] \) is equal to
(a) 8
(b) 16
(c) 64
(d) None of the options
Answer: (b) 16

Question. If \( \vec{d} \) is a unit vector such that \( \vec{d} = \lambda(\vec{b} \times \vec{c}) + \mu(\vec{c} \times \vec{a}) + v(\vec{a} \times \vec{b}) \) then \( |(\vec{d} \cdot \vec{a})(\vec{b} \times \vec{c}) + (\vec{d} \cdot \vec{b})(\vec{c} \times \vec{a}) + (\vec{d} \cdot \vec{c})(\vec{a} \times \vec{b})| \) is equal to
(a) \( |[\vec{a} \vec{b} \vec{c}]| \)
(b) 1
(c) \( 3|[\vec{a} \vec{b} \vec{c}]| \)
(d) None of the options
Answer: (a) \( |[\vec{a} \vec{b} \vec{c}]| \)

Question. \( \vec{a} \times (\vec{b} \times \vec{c}), \vec{b} \times (\vec{c} \times \vec{a}) \) and \( \vec{c} \times (\vec{a} \times \vec{b}) \) are
(a) linearly
(b) dependent
(c) equal vectors
(d) None of the options
Answer: (a) linearly

Question. \( [\vec{b} \vec{c} \quad \vec{b} \times \vec{c}] + (\vec{b} \cdot \vec{c})^2 \) is equal to
(a) \( |\vec{b}|^2|\vec{c}|^2 \)
(b) \( (\vec{b} + \vec{c})^2 \)
(c) \( |\vec{b}|^2 + |\vec{c}|^2 \)
(d) None of the options
Answer: (a) \( |\vec{b}|^2|\vec{c}|^2 \)

Question. If the vector \( \vec{a}, \vec{b}, \vec{c} \) and \( \vec{d} \) are coplanar then \( (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \) is equal to
(a) \( \vec{a} + \vec{b} + \vec{c} + \vec{d} \)
(b) \( \vec{0} \)
(c) \( \vec{a} + \vec{b} = \vec{c} + \vec{d} \)
(d) None of the options
Answer: (b) \( \vec{0} \)

Question. If \( \vec{a} \times (\vec{a} \times \vec{b}) = \vec{b} \times (\vec{b} \times \vec{c}) \) and \( \vec{a} \cdot \vec{b} \neq 0 \) then \( [\vec{a} \vec{b} \vec{c}] \) is equal to
(a) 0
(b) 1
(c) 2
(d) None of the options
Answer: (a) 0

Question. Let \( \vec{OA} = \vec{a}, \vec{OB} = 10\vec{a} + 2\vec{b} \) and \( \vec{OC} = \vec{b} \), where O, A and C noncollinear points. Let p denote the area of the quadrilateral OAB, and q denote the area of the parallelogram with OA and OC as adjacent sides. Then p/q is equal to
(a) 4
(b) 6
(c) \( \frac{1}{2} \frac{|\vec{a} - \vec{b}|}{|\vec{a}|} \)
(d) None of the options
Answer: (b) 6

Question. The position vectors of the vertices A, B, C of a triangle are \( \vec{i} - \vec{j} - 3\vec{k}, 2\vec{i} + \vec{j} - 2\vec{k} \) and \( -5\vec{i} + 2\vec{j} - 6\vec{k} \) respectively. The length of the bisector AD of the angle BAC where D is on the line segment BC, is
(a) \( \frac{15}{2} \)
(b) \( \frac{1}{4} \)
(c) \( \frac{11}{2} \)
(d) None of the options
Answer: (a) \( \frac{15}{2} \)

Question. P is a point on the line through the point A whose position vector is \( \vec{a} \) and the line is parallel to the vector \( \vec{b} \). If PA = 6, the position vector of P is
(a) \( \vec{a} + 6\vec{b} \)
(b) \( \vec{a} + \frac{6}{|\vec{b}|}\vec{b} \)
(c) \( \vec{a} - 6\vec{b} \)
(d) \( \vec{b} + \frac{6}{|\vec{a}|}\vec{a} \)
Answer: (b) \( \vec{a} + \frac{6}{|\vec{b}|}\vec{b} \)

Question. The coplanar points A,B,C,D are (2 – x, 2, 2), (2, 2 – y, 2), (2, 2, 2 – z) and (1, 1, 1) respectively. Then
(a) \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 \)
(b) x + y + z = 1
(c) \( \frac{1}{1-x} + \frac{1}{1-y} + \frac{1}{1-z} \)
(d) None of the options
Answer: (a) \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 \)

Question. Let \( \vec{AB} = 3\vec{i} + \vec{j} - \vec{k} \) and \( \vec{AC} = \vec{i} - \vec{j} + 3\vec{k} \). If the point P on the line segment BC is equidistant from AB and AC then \( \vec{AP} \) is
(a) \( 2\vec{i} - \vec{k} \)
(b) \( \vec{i} - 2\vec{k} \)
(c) \( 2\vec{i} + \vec{k} \)
(d) None of the options
Answer: (c) \( 2\vec{i} + \vec{k} \)

Question. The cosine of the angle between two diagonals of a cube is
(a) \( \frac{1}{3} \)
(b) \( \frac{2\sqrt{2}}{3} \)
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (a) \( \frac{1}{3} \)

Question. If \( \vec{AB} = \vec{b} \) and \( \vec{AC} = \vec{c} \) then the length of the perpendicular from A to the line BC is
(a) \( \frac{|\vec{b} \times \vec{c}|}{|\vec{b} + \vec{c}|} \)
(b) \( \frac{|\vec{b} \times \vec{c}|}{|\vec{b} - \vec{c}|} \)
(c) \( \frac{1}{2} \frac{|\vec{b} \times \vec{c}|}{|\vec{b} - \vec{c}|} \)
(d) None of the options
Answer: (b) \( \frac{|\vec{b} \times \vec{c}|}{|\vec{b} - \vec{c}|} \)

Question. The distance of the point (1, 1, 1) from the plane passing through the points (2, 1, 1), (1, 2, 1) and (1, 1, 2) is
(a) \( \frac{1}{\sqrt{3}} \)
(b) 1
(c) \( \sqrt{3} \)
(d) None of the options
Answer: (a) \( \frac{1}{\sqrt{3}} \)

Question. The projection of the vector \( \vec{i} + \vec{j} + \vec{k} \) on the line whose vector equation is \( \vec{r} = (3 + t)\vec{i} + (2t - 1)\vec{j} + 3t\vec{k} \), t being the scalar parameter, is
(a) \( \frac{1}{\sqrt{14}} \)
(b) 6
(c) \( \frac{6}{\sqrt{14}} \)
(d) None of the options
Answer: (c) \( \frac{6}{\sqrt{14}} \)

Question. If the vertices of a tetrahedron have the position vectors \( \vec{0}, \vec{i} + \vec{j}, 2\vec{j} - \vec{k} \) and \( \vec{i} + \vec{k} \) then the volume of the volume if of the tetrahedron is
(a) \( \frac{1}{6} \)
(b) 1
(c) 2
(d) None of the options
Answer: (a) \( \frac{1}{6} \)

Question. A line passes through the point whose position vectors are \( \vec{i} + \vec{j} + 2\vec{k} \) and \( \vec{i} - 3\vec{j} + \vec{k} \). The position vector of a point on it at a unit distance from the first point is
(a) \( \frac{1}{5}(6\vec{i} + \vec{j} - 7\vec{k}) \)
(b) \( \frac{1}{5}(4\vec{i} + 9\vec{j} - 13\vec{k}) \)
(c) \( \vec{i} - 4\vec{j} + 3\vec{k} \)
(d) None of the options
Answer: (a), (b)

Question. A vector of magnitude 2 along bisector of the angle between the two vectors \( 2\vec{i} - 2\vec{j} + \vec{k} \) and \( \vec{i} + 2\vec{j} - 2\vec{k} \) is
(a) \( \frac{2}{\sqrt{10}}(\vec{i} - \vec{k}) \)
(b) \( \frac{1}{\sqrt{26}}(\vec{i} - 4\vec{j} + 3\vec{k}) \)
(c) \( \frac{2}{\sqrt{26}}(\vec{i} - 4\vec{j} + 3\vec{k}) \)
(d) None of the options
Answer: (a), (c)

Question. A unit vector coplanar with \( \vec{i} + \vec{j} + 2\vec{k} \) and \( \vec{i} + 2\vec{j} + \vec{k} \), and perpendicular to \( \vec{i} + \vec{j} + \vec{k} \), is
(a) \( \frac{1}{\sqrt{2}}(-\vec{j} + \vec{k}) \)
(b) \( \frac{1}{\sqrt{2}}(\vec{k} - \vec{i}) \)
(c) \( \frac{1}{\sqrt{2}}(\vec{i} - \vec{k}) \)
(d) \( \frac{1}{\sqrt{2}}(\vec{j} - \vec{k}) \)
Answer: (a), (d)

Question. A unit vector which is equally inclined to the vectors \( \vec{i}, \frac{-2\vec{i} + \vec{j} + 2\vec{k}}{3} \) and \( \frac{-4\vec{j} - 3\vec{k}}{5} \) is
(a) \( \frac{1}{\sqrt{51}}(-\vec{i} + 5\vec{j} - 5\vec{k}) \)
(b) \( \frac{1}{\sqrt{51}}(\vec{i} + 5\vec{j} - 5\vec{k}) \)
(c) \( \frac{1}{\sqrt{51}}(\vec{i} + 5\vec{j} + 5\vec{k}) \)
(d) \( \frac{1}{\sqrt{51}}(-\vec{i} + 5\vec{j} + 5\vec{k}) \)
Answer: (a), (d)

Question. If \( |\vec{a}| = 4, |\vec{b}| = 2 \) and the angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{6} \) then \( (\vec{a} \times \vec{b})^2 \) equal to
(a) 48
(b) 16
(c) \( \vec{a}^2 \)
(d) None of the options
Answer: (b), (c)

Question. Three points whose position vectors are \( \vec{a}, \vec{b}, \vec{c} \) will be collinear if
(a) \( \lambda\vec{a} + \mu\vec{b} = (\lambda + \mu)\vec{c} \)
(b) \( \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 0 \)
(c) \( [\vec{a} \vec{b} \vec{c}] \)
(d) None of the options
Answer: (a), (b)

Question. Let \( \vec{b} = 4\vec{i} + 3\vec{j} \). Let \( \vec{c} \) be a vector perpendicular to \( \vec{b} \) and it lies in the x–y plane. A vector in the x–y plane having projections 1 and 2 along \( \vec{b} \) and \( \vec{c} \) is
(a) \( 2\vec{i} - \vec{j} \)
(b) \( \vec{i} - 2\vec{j} \)
(c) \( \frac{1}{5}(-\vec{i} + 11\vec{j}) \)
(d) None of the options
Answer: (a), (c)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are noncoplanar nonzero vectors and \( \vec{r} \) is any vector in space then \( [\vec{b} \vec{c} \vec{r}]\vec{a} + [\vec{c} \vec{a} \vec{r}]\vec{b} + [\vec{a} \vec{b} \vec{r}]\vec{c} \) is equal to
(a) \( 3[\vec{a} \vec{b} \vec{c}]\vec{r} \)
(b) \( [\vec{a} \vec{b} \vec{c}]\vec{r} \)
(c) \( [\vec{b} \vec{c} \vec{a}]\vec{r} \)
(d) None of the options
Answer: (b), (c)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are nonecoplanar vectors such that \( \vec{b} \times \vec{c} = \vec{a}, \vec{a} \times \vec{b} = \vec{c} \) and \( \vec{c} \times \vec{a} = \vec{b} \) then
(a) \( |\vec{a}| = 1 \)
(b) \( |\vec{b}| = 1 \)
(c) \( |\vec{a}| + |\vec{b}| + |\vec{c}| = 3 \)
(d) None of the options
Answer: (a), (b), (c)

Question. Let \( \vec{a}, \vec{b}, \vec{c} \) be noncoplanar vectors and \( \vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{b} \vec{c} \vec{a}]}, \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{c} \vec{a} \vec{b}]} \). Then
(a) \( \vec{p} \cdot \vec{a} = 1 \)
(b) \( \vec{p} \cdot \vec{a} + \vec{q} \cdot \vec{b} + \vec{r} \cdot \vec{c} = 3 \)
(c) \( \vec{p} \cdot \vec{a} + \vec{q} \cdot \vec{b} + \vec{r} \cdot \vec{c} = 0 \)
(d) None of the options
Answer: (a), (b)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are any three vectors then \( (\vec{a} \times \vec{b}) \times \vec{c} \) is a vector
(a) perpendicular to \( \vec{a} \times \vec{b} \)
(b) coplanar with \( \vec{a} \) and \( \vec{b} \)
(c) parallel to \( \vec{c} \)
(d) parallel to either \( \vec{a} \) or \( \vec{b} \)
Answer: (a), (b)

Question. If \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{b} = \vec{c} \times \vec{a} \) then
(a) \( \vec{a} \cdot \vec{b} = c^2 \)
(b) \( \vec{c} \cdot \vec{a} = b \)
(c) \( \vec{a} \perp \vec{b} \)
(d) \( \vec{a} \parallel \vec{b} \times \vec{c} \)
Answer: (c), (d)

Question. If \( \vec{x} \times \vec{b} = \vec{c} \times \vec{b} \) and \( \vec{x} \perp \vec{a} \) then \( \vec{x} \) is equal to
(a) \( \frac{\vec{b} \times (\vec{a} \times \vec{c})}{\vec{b} \cdot \vec{c}} \)
(b) \( \frac{(\vec{b} \times \vec{c}) \times \vec{a}}{\vec{b} \cdot \vec{a}} \)
(c) \( \frac{\vec{a} \times (\vec{c} \times \vec{b})}{\vec{a} \cdot \vec{b}} \)
(d) None of the options
Answer: (b), (c)

Question. The resolved part of the vector \( \vec{a} \) along the vector \( \vec{b} \) is \( \vec{\lambda} \) and that perpendicular to \( \vec{b} \) is \( \vec{\mu} \). Then
(a) \( \vec{\lambda} = \frac{(\vec{a} \cdot \vec{b})\vec{a}}{a^2} \)
(b) \( \vec{\lambda} = \frac{(\vec{a} \cdot \vec{b})\vec{b}}{b^2} \)
(c) \( \vec{\mu} = \frac{(\vec{b} \cdot \vec{b})\vec{a} - (\vec{a} \cdot \vec{b})\vec{b}}{b^2} \)
(d) \( \vec{\mu} = \frac{\vec{b} \times (\vec{a} \times \vec{b})}{b^2} \)
Answer: (b), (c), (d)

Question. \( (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) \) is equal to
(a) \( \vec{a} \cdot \{ \vec{b} \times (\vec{c} \times \vec{d}) \} \)
(b) \( (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c}) \)
(c) \( \{(\vec{a} \times \vec{b}) \times \vec{c}\} \cdot \vec{d} \)
(d) \( (\vec{d} \times \vec{c}) \cdot (\vec{b} \times \vec{a}) \)
Answer: (a), (b), (c), (d)

Question. If \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) are any four then \( (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \) is a vactor
(a) perpendicular to \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \)
(b) along the line of intersection of two planes, one containing \( \vec{a}, \vec{b} \) and the other containing \( \vec{c}, \vec{d} \)
(c) equally inclined to both \( \vec{a} \times \vec{b} \) and \( \vec{c} \times \vec{d} \)
(d) None of the options
Answer: (b), (c)