JEE Mathematics Vectors and Their Applications MCQs Set 02

Question. Then nit vector perpendicular to both the vectors \( \vec{a} = \vec{i} + \vec{j} + \vec{k} \) and \( \vec{b} = 2\vec{i} - \vec{j} + 3\vec{k} \) and making an acute angle with the vector \( \vec{k} \) is
(a) \( -\frac{1}{\sqrt{26}}(4\vec{i} - \vec{j} - 3\vec{k}) \)
(b) \( \frac{1}{\sqrt{26}}(4\vec{i} - \vec{j} - 3\vec{k}) \)
(c) \( \frac{1}{\sqrt{26}}(4\vec{i} - \vec{j} + 3\vec{k}) \)
(d) None of the options
Answer: (a) \( -\frac{1}{\sqrt{26}}(4\vec{i} - \vec{j} - 3\vec{k}) \)

Question. Let \( \vec{a} = \vec{i} - 2\vec{j} + 3\vec{k}, \vec{b} = 2\vec{i} + 3\vec{j} - \vec{k} \) and \( \vec{c} = \lambda\vec{i} + (2\lambda - 1)\vec{k} \). If \( \vec{c} \) is parallel to the plane of the vectors \( \vec{a} \) and \( \vec{b} \) then \( \lambda \) is
(a) \( 1 \)
(b) \( 0 \)
(c) \( -1 \)
(d) \( 2 \)
Answer: (b) \( 0 \)

Question. Let \( \vec{a} \) be a unit vector perpendicular to unit vectors \( \vec{b} \) and \( \vec{c} \) and if the angle between \( \vec{b} \) and \( \vec{c} \) be \( \alpha \) then \( \vec{b} \times \vec{c} \) is
(a) \( \cos \alpha \vec{a} \)
(b) \( \text{cosec } \alpha \vec{a} \)
(c) \( \sin \alpha \vec{a} \)
(d) None of the options
Answer: (c) \( \sin \alpha \vec{a} \)

Question. If \( \vec{a} \cdot \vec{b} = 0 \) and \( \vec{a} \times \vec{b} = \vec{0} \) then
(a) \( \vec{a} \parallel \vec{b} \)
(b) \( \vec{a} \perp \vec{b} \)
(c) \( \vec{a} = \vec{0} \) or \( \vec{b} = \vec{0} \)
(d) None of the options
Answer: (c) \( \vec{a} = \vec{0} \) or \( \vec{b} = \vec{0} \)

Question. The area of the parallelogram whose diagonals represent the vectors \( 3\vec{i} + \vec{j} - 2\vec{k} \) and \( \vec{i} - 3\vec{j} + 4\vec{k} \) is
(a) \( 10\sqrt{3} \)
(b) \( 5\sqrt{3} \)
(c) \( 8 \)
(d) \( 4 \)
Answer: (b) \( 5\sqrt{3} \)

Question. \( (\vec{r} \cdot \vec{i})(\vec{r} \times \vec{i}) + (\vec{r} \cdot \vec{j})(\vec{r} \times \vec{j}) + (\vec{r} \cdot \vec{k})(\vec{r} \times \vec{k}) \) is equal to
(a) \( 3\vec{r} \)
(b) \( \vec{r} \)
(c) \( 8 \)
(d) None of the options
Answer: (c) \( 8 \)

Question. Let \( \vec{a} = \vec{i} + \vec{j} + \vec{k}, \vec{c} = \vec{j} - \vec{k} \). If \( \vec{b} \) is a vector satisfying \( \vec{a} \times \vec{b} = \vec{c} \) and \( \vec{a} \cdot \vec{b} = 3 \) then \( \vec{b} \) is
(a) \( \frac{1}{3}(5\vec{i} + 2\vec{j} + 2\vec{k}) \)
(b) \( \frac{1}{3}(5\vec{i} - 2\vec{j} - 2\vec{k}) \)
(c) \( 3\vec{i} - \vec{j} - \vec{k} \)
(d) None of the options
Answer: (a) \( \frac{1}{3}(5\vec{i} + 2\vec{j} + 2\vec{k}) \)

Question. A unit vector perpendicular to the plane passing through the points whose position vector are \( \vec{i} - \vec{j} + 2\vec{k}, 2\vec{i} - \vec{k} \) and \( 2\vec{i} + \vec{k} \) is
(a) \( 2\vec{i} + \vec{j} + \vec{k} \)
(b) \( \frac{1}{\sqrt{6}}(2\vec{i} + \vec{j} + \vec{k}) \)
(c) \( \frac{1}{\sqrt{6}}(\vec{i} + 2\vec{j} + \vec{k}) \)
(d) None of the options
Answer: (b) \( \frac{1}{\sqrt{6}}(2\vec{i} + \vec{j} + \vec{k}) \)

Question. Let \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \) and v, where \( \vec{a} \cdot \vec{b} \neq 0 \). Then \( \vec{r} \) is equal to
(a) \( \vec{b} + t\vec{a} \) where t is a scalar
(b) \( \vec{b} - \frac{\vec{b} \cdot \vec{c}}{\vec{a} \cdot \vec{c}}\vec{a} \)
(c) \( \vec{a} - \vec{c} \)
(d) None of the options
Answer: (b) \( \vec{b} - \frac{\vec{b} \cdot \vec{c}}{\vec{a} \cdot \vec{c}}\vec{a} \)

Question. For the vectors \( \vec{u}, \vec{v}, \vec{w} \) which of the following expressions is not equal to any one of the remaining three option?
(a) \( \vec{u} \cdot (\vec{v} \times \vec{w}) \)
(b) \( (\vec{v} \times \vec{w}) \cdot \vec{u} \)
(c) \( \vec{v} \cdot (\vec{u} \times \vec{w}) \)
(d) \( (\vec{u} \times \vec{v}) \cdot \vec{w} \)
Answer: (c) \( \vec{v} \cdot (\vec{u} \times \vec{w}) \)

Question. For three noncoplanar vectors \( \vec{a}, \vec{b}, \vec{c} \) the relation hold \( |\vec{a} \times \vec{b} \cdot \vec{c}| = |\vec{a}| |\vec{b}| |\vec{c}| \) holds if and only if
(a) \( \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0 \)
(b) \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = 0 \)
(c) \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0 \)
(d) \( \vec{c} \cdot \vec{a} = \vec{a} \cdot \vec{b} = 0 \)
Answer: (c) \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0 \)

Question. \( [\vec{a} + \vec{b} \quad \vec{b} + \vec{c} \quad \vec{c} + \vec{a}] \) is equal to
(a) \( 2[\vec{a} \vec{b} \vec{c}] \)
(b) \( 3[\vec{a} \vec{b} \vec{c}] \)
(c) \( [\vec{a} \vec{b} \vec{c}] \)
(d) 0
Answer: (a) \( 2[\vec{a} \vec{b} \vec{c}] \)

Question. \( \vec{a} - \vec{b} \quad \vec{b} - \vec{c} \quad \vec{c} - \vec{a} \) is equal to
(a) \( 2[\vec{a} \vec{b} \vec{c}] \)
(b) \( [\vec{a} \vec{b} \vec{c}] \)
(c) 0
(d) None of the options
Answer: (c) 0

Question. Let \( \vec{a}, \vec{b}, \vec{c} \) be three unit vectors and \( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0 \). If the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{3} \) then \( |[\vec{a} \vec{b} \vec{c}]| \) is equal to
(a) \( \frac{\sqrt{3}}{2} \)
(b) \( \frac{1}{2} \)
(c) 1
(d) None of the options
Answer: (a) \( \frac{\sqrt{3}}{2} \)

Question. Let a, b, c, be three distinct positive real numbers. If \( \vec{p}, \vec{q}, \vec{r} \) lie in a plane, where \( \vec{p} = a\vec{i} - a\vec{j} + b\vec{k}, \vec{q} = \vec{i} + \vec{k} \) and \( \vec{r} = c\vec{i} + c\vec{j} + b\vec{k} \), then b is
(a) then AM of a, c
(b) then GM if a, c
(c) then HM of a, c
(d) equal to 0
Answer: (c) then HM of a, c

Question. Which of the following is not equal to \( [\vec{a} \vec{b} \vec{c}] \)?
(a) \( \vec{a} \cdot \vec{b} \times \vec{c} \)
(b) \( \vec{c} \times \vec{a} \cdot \vec{b} \)
(c) \( \vec{b} \cdot \vec{a} \times \vec{c} \)
(d) \( \vec{c} \cdot \vec{a} \times \vec{b} \)
Answer: (c) \( \vec{b} \cdot \vec{a} \times \vec{c} \)

Question. If \( [\vec{a} \vec{b} \vec{c}] = 1 \) then \( \frac{\vec{a} \cdot \vec{b} \times \vec{c}}{\vec{c} \times \vec{a} \cdot \vec{b}} + \frac{\vec{b} \cdot \vec{c} \times \vec{a}}{\vec{a} \times \vec{b} \cdot \vec{c}} + \frac{\vec{c} \cdot \vec{a} \times \vec{b}}{\vec{b} \times \vec{c} \cdot \vec{a}} \) is equal to
(a) 3
(b) 1
(c) 0
(d) None of the options
Answer: (a) 3

Question. \( \vec{a}, \vec{b}, \vec{c} \) are noncoplanar vectors and \( \vec{p}, \vec{q}, \vec{r} \) are defined as \( \vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{b} \vec{c} \vec{a}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{c} \vec{a} \vec{b}]}, \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]} \). Then \( (\vec{a} + \vec{b}) \cdot \vec{p} + (\vec{b} + \vec{c}) \cdot \vec{q} + (\vec{c} + \vec{a}) \cdot \vec{r} \) is equal to
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (d) 3

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are three noncoplanar vectors represented by concurrent edges of a parallelepiped of volume 4 then \( (\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} + \vec{a}) \cdot (\vec{a} \times \vec{b}) \) is equal to
(a) 12
(b) 4
(c) \( \pm 12 \)
(d) 0
Answer: (c) \( \pm 12 \)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are three noncoplanar nonzero vectors then \( (\vec{a} \cdot \vec{a})\vec{b} \times \vec{c} + (\vec{a} \cdot \vec{b})\vec{c} \times \vec{a} + (\vec{a} \cdot \vec{c})\vec{a} \times \vec{b} \) is equal to
(a) \( [\vec{b} \vec{c} \vec{a}]\vec{a} \)
(b) \( [\vec{c} \vec{a} \vec{b}]\vec{b} \)
(c) \( [\vec{a} \vec{b} \vec{c}]\vec{c} \)
(d) None of the options
Answer: (a) \( [\vec{b} \vec{c} \vec{a}]\vec{a} \)

Question. Let \( \vec{r} \) be a vector perpendicular to \( \vec{a} + \vec{b} + \vec{c} \), where \( [\vec{a} \vec{b} \vec{c}] = 2 \). If \( \vec{r} = l(\vec{b} \times \vec{c}) + m(\vec{c} \times \vec{a}) + n(\vec{a} \times \vec{b}) \) then l + m + n is
(a) 2
(b) 1
(c) 0
(d) None of the options
Answer: (c) 0

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are any three vectors in space then \( (\vec{c} + \vec{b}) \times (\vec{c} + \vec{a}) \cdot (\vec{c} + \vec{b} + \vec{a}) \) is equal to
(a) \( 3[\vec{a} \vec{b} \vec{c}] \)
(b) 0
(c) \( [\vec{a} \vec{b} \vec{c}] \)
(d) None of the options
Answer: (c) \( [\vec{a} \vec{b} \vec{c}] \)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are three noncoplanar vectors then \( [\vec{a} + \vec{b} + \vec{c} \quad \vec{a} - \vec{c} \quad \vec{a} - \vec{b}] \) is equal to
(a) 0
(b) \( [\vec{a} \vec{b} \vec{c}] \)
(c) \( -3[\vec{a} \vec{b} \vec{c}] \)
(d) \( 2[\vec{a} \vec{b} \vec{c}] \)
Answer: (c) \( -3[\vec{a} \vec{b} \vec{c}] \)

Question. \( [\vec{a} + \vec{b} \quad \vec{b} + \vec{c} \quad \vec{a} + \vec{b} + \vec{c}] \) is equal to
(a) 0
(b) \( 2[\vec{a} \vec{b} \vec{c}] \)
(c) \( [\vec{a} \vec{b} \vec{c}] \)
(d) None of the options
Answer: (a) 0

Question. If \( \vec{a}, \vec{b} \) are nonezero and noncollinear vectors then \( [\vec{a}, \vec{b}, \vec{i}]\vec{i} + [\vec{a}, \vec{b}, \vec{j}]\vec{j} + [\vec{a}, \vec{b}, \vec{k}]\vec{k} \) is equal to
(a) \( \vec{a} + \vec{b} \)
(b) \( \vec{a} \times \vec{b} \)
(c) \( \vec{a} - \vec{b} \)
(d) \( \vec{b} \times \vec{a} \)
Answer: (b) \( \vec{a} \times \vec{b} \)

Question. The three concurrent edges of a parallelepiped represent the vectors \( \vec{a}, \vec{b}, \vec{c} \) such that \( [\vec{a} \vec{b} \vec{c}] = \lambda \). Then the volume of the parallelepiped whose three concurrent edges are the three concurrent diagonals of three faces of the given parallelepiped is
(a) \( 2\lambda \)
(b) \( 3\lambda \)
(c) \( \lambda \)
(d) None of the options
Answer: (a) \( 2\lambda \)

Question. \( \vec{i} \times (\vec{a} \times \vec{i}) + \vec{j} \times (\vec{a} \times \vec{j}) + \vec{k} \times (\vec{a} \times \vec{k}) \) is equal to
(a) \( 2\vec{a} \)
(b) \( 3\vec{a} \)
(c) \( \vec{0} \)
(d) None of the options
Answer: (a) \( 2\vec{a} \)

Question. Let \( \vec{a}, \vec{b} \) and \( \vec{c} \) be three vectors having magnitudes 1, 1 and 2 respectively. If \( \vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0} \), the acute angle between \( \vec{a} \) and \( \vec{c} \) is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{6} \)
(d) None of the options
Answer: (c) \( \frac{\pi}{6} \)

Question. If \( \vec{b} \) is a unit vector then \( (\vec{a} \cdot \vec{b})\vec{b} + \vec{b} \times (\vec{a} \times \vec{b}) \) is equal to
(a) \( (\vec{a} \cdot \vec{b})^2 \)
(b) \( (\vec{a} \cdot \vec{b})\vec{a} \)
(c) \( \vec{a} \)
(d) None of the options
Answer: (c) \( \vec{a} \)

Question. Let \( \vec{a}, \vec{b}, \vec{c} \) be three unit vectors such that \( \vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} + \vec{c}}{\sqrt{2}} \) and the angles between \( \vec{a}, \vec{c} \) and \( \vec{a}, \vec{b} \) be \( \alpha \) and \( \beta \) respectively then
(a) \( \alpha = \frac{3\pi}{4}, \beta = \frac{\pi}{4} \)
(b) \( \alpha = \frac{\pi}{4}, \beta = \frac{7\pi}{4} \)
(c) \( \alpha = \frac{\pi}{4}, \beta = \frac{3\pi}{4} \)
(d) None of the options
Answer: (c) \( \alpha = \frac{\pi}{4}, \beta = \frac{3\pi}{4} \)

Question. Let \( \vec{p}, \vec{q}, \vec{r} \) be three mutually perpendicular vectors of the same magnitude. If a vector \( \vec{x} \) satisfies the equation \( \vec{p} \times \{(\vec{x} - \vec{q}) \times \vec{p}\} + \vec{q} \times \{(\vec{x} - \vec{r}) \times \vec{q}\} + \vec{r} \times \{(\vec{x} - \vec{p}) \times \vec{r}\} = \vec{0} \) this x is given by
(a) \( \frac{1}{2}(\vec{p} + \vec{q} - 2\vec{r}) \)
(b) \( \frac{1}{2}(\vec{p} + \vec{q} + \vec{r}) \)
(c) \( \frac{1}{3}(\vec{p} + \vec{q} + \vec{r}) \)
(d) \( \frac{1}{3}(2\vec{p} + \vec{q} + \vec{r}) \)
Answer: (b) \( \frac{1}{2}(\vec{p} + \vec{q} + \vec{r}) \)

Question. If . and x represent dot product and cross product respectively then which of the following is meaningless?
(a) \( (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) \)
(b) \( (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \)
(c) \( (\vec{a} \cdot \vec{b})(\vec{c} \times \vec{d}) \)
(d) \( (\vec{a} \cdot \vec{b}) \times (\vec{c} \times \vec{d}) \)
Answer: (d) \( (\vec{a} \cdot \vec{b}) \times (\vec{c} \times \vec{d}) \)

Question. \( (\vec{a} \times \vec{i})^2 + (\vec{a} \times \vec{j})^2 + (\vec{a} \times \vec{k})^2 \) is equal to
(a) \( \vec{a}^2 \)
(b) \( 3\vec{a}^2 \)
(c) \( 2\vec{a}^2 \)
(d) None of the options
Answer: (c) \( 2\vec{a}^2 \)

Question. If \( \vec{a} \parallel \vec{b} \times \vec{c} \) then \( (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c}) \) is equal to
(a) \( \vec{a}^2(\vec{b} \cdot \vec{c}) \)
(b) \( \vec{b}^2(\vec{a} \cdot \vec{c}) \)
(c) \( \vec{c}^2(\vec{a} \cdot \vec{b}) \)
(d) None of the options
Answer: (a) \( \vec{a}^2(\vec{b} \cdot \vec{c}) \)

Question. If \( \vec{a}, \vec{b}, \vec{c} \) are noncoplanar nonzero vectors then \( (\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c}) + (\vec{b} \times \vec{c}) \times (\vec{b} \times \vec{a}) + (\vec{c} \times \vec{a}) \times (\vec{c} \times \vec{b}) \) is equal to
(a) \( [\vec{a} \vec{b} \vec{c}]^2(\vec{a} + \vec{b} + \vec{c}) \)
(b) \( [\vec{a} \vec{b} \vec{c}](\vec{a} + \vec{b} + \vec{c}) \)
(c) \( \vec{0} \)
(d) None of the options
Answer: (b) \( [\vec{a} \vec{b} \vec{c}](\vec{a} + \vec{b} + \vec{c}) \)