Chapter 21

Let \(V(x, y, z)=V_{1} i+V_{2} j+V_{3} k\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the curl or roation of \(\vec{V}\), written \(\nabla \times \vec{V}\), curl \(\vec{V}\) or rot \(\vec{V}\), is defined by \(\begin{aligned} \nabla \times V &=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \times\left(V_{1} i+V_{2} j+V_{3} k\right) \\ &=\left|\begin{array}{lll}i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ V_{1} & V_{2} & V_{3}\end{array}\right| \\\ &=\left(\frac{\partial V_{3}}{\partial y}-\frac{\partial V_{2}}{\partial z_{2}}\right) i+\left(\frac{\partial V_{2}}{\partial z}-\frac{\partial V_{3}}{\partial x}\right) j+\left(\frac{\partial V_{2}}{\partial x}-\frac{\partial V_{1}}{\partial y}\right) k \end{aligned}\) Note that in the expansion of the determinant the operators \(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\) must precede \(V_{1}, V_{2}, V_{3}\). For a vector function \(A\) possessing continuous second order partial derivatives, \(\nabla \cdot(\nabla \times A)=\) (A) \(A\) (B) \(\nabla \times A\) (C) 0 (D) none of these

Column-I I. The points \(O, A, B, C, D\) are such that \(O A=a, O B=b, O C=2 a+3 b\) and \(O D\) \(=a-2 b\). If \(|a|=3|b|\), then the angle between \(B D\) and \(A C\) is II. \(a, b, c\) are three unit vectors such that \(a \times(b \times c)=\frac{1}{2}(b+c) .\) If the vectors \(b\) and \(c\) are non-parallel, then the angle between \(a\) and \(b\) is III. Let the vectors \(a, b, c\) and \(d\) be such that \((a \times b) \times(c \times d)=0\). Let \(P_{1}\) and \(P_{2}\) be planes determined by the pairs of vectors \(a, b\) and \(c, d\) respectively, then the angle between \(P_{1}\) and \(P_{2}\) is IV. If \(a\) and \(b\) are two vectors such that \(a \cdot b<0\) and \(|a \cdot b|=|a \times b|\), then the angle between vectors \(a\) and \(b\) is Column-I (A) 0 (B) \(\frac{2 \pi}{3}\) (C) \(\frac{3 \pi}{4}\) (D) \(\frac{\pi}{2}\)

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(i, j, k\) are orthonormal unit vectors and \(a\) is any vector. If \(a \times r=j=\hat{j}\), then \(a \cdot r\) is arbitrary scalar. Reason: For any two arbitrary vectors \(a\) and \(r\), \(|a \times r|^{2}+|a \cdot r|^{2}=|a|^{2}|r|^{2}\)

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If \(a, b, c\) are three non-coplanar, non-zero vectors, then \((a \cdot a) b \times c+(a \cdot b) c \times a+(a \cdot c) a \times b=[b c a] a\) Reason: If the vectors \(a, b, c\) are non-coplanar, then so are \(b \times c, c \times a, a \times b\)

In the following questions an Assertion (A) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If $$ \begin{aligned} &i \times[(a-j) \times i]+j \times[(a-k) \times j]+k \\ &\times[(a-i) \times k]=0, \text { then } \\ &a=\frac{1}{2}(i+j+k) \end{aligned} $$ Reason: \((a \cdot i) i+(a \cdot j) j+(a \cdot k) k=a\) for any vector \(a\).

Given two vectors are \(\hat{i}-\hat{j}\) and \(\hat{i}+2 \hat{j}\) the unit vector coplanar with the two vectors and perpendicular to first is: (A) \(\frac{\underline{\phantom{xx}}}{\sqrt{t}}(\hat{i}\) \(\hat{j})\) (B) \(\frac{1}{\sqrt{5}}(2 \hat{i}+\hat{j})\) (C) \(\pm \frac{1}{\sqrt{2}}(\hat{i}+\hat{k})\) (D) none of these

If \(\left|\begin{array}{lll}a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0\) and vectors \(\left(1, a, a^{2}\right)\left(1, b, b^{2}\right)\) and \(\left(1, c, c^{2}\right)\) are non-coplanar, then the product \(a b c\) equals [2003] (A) 2 (B) \(-1\) (C) 1 (D) 0

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that no two of these are collinear. If the vector \(\vec{a}+2 \vec{b}\) is collinear with \(\vec{c}\) and \(\vec{b}+3 \vec{c}\) is collinear with \(\vec{a}\) ( \(\lambda\) being some non-zero scalar) then \(\vec{a}+2 \vec{b}+6 \vec{c}\) equals [2004] (A) \(\lambda \vec{a}\) (B) \(\lambda \vec{b}\) (C) \(\lambda \vec{c}\) (D) 0

A particle is acted upon by constant forces \(4 \hat{i}+\hat{j}-3 \hat{k}\) and \(3 \hat{i}+\hat{j}-\hat{k}\) which displace it from a point \(\hat{i}+2 \hat{j}+3 \hat{k}\) to the point \(5 \hat{i}+4 \hat{j}+\hat{k}\). The work done in standard units by the forces is given by (A) 40 (B) 30 (C) 25 (D) 15

If \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors \(\bar{a}+2 \bar{b}+3 \bar{c}, \lambda \bar{b}+4 \bar{c}\) and \((2 \lambda-1) \bar{c}\) are non-coplanar for (A) all values of \(\lambda\) (B) all except one value of \(\lambda\) (C) all except two values of \(\lambda\) (D) no value or \(\lambda\)

Let \(\bar{u}, \bar{v}, \bar{w}\) be such that \(|\bar{u}|=1,|\bar{v}|=2,|\bar{w}|=3\). If the projection \(\bar{v}\) along \(\bar{u}\) is equal to that of \(\bar{w}\) along \(\bar{u}\) and \(\bar{v}, \bar{w}\) are perpendicular to each other then \(\bar{u}-\bar{v}+\bar{w} \mid\) equals (A) 2 (B) \(\sqrt{7}\) (C) \(\sqrt{14}\) (D) 14

Let \(\bar{a}, \bar{b}\) and \(\bar{c}\) be non-zero vectors such that \((\bar{a} \times \bar{b}) \times \bar{c}=-|\bar{b}| \bar{c} \mid \bar{a} . \quad\) If \(\theta\) is the acute angle between the vectors \(\bar{b}\) and \(\bar{c}\) then \(\sin \theta\) equals [2004] (A) \(\frac{1}{3}\) (B) \(\frac{\sqrt{2}}{3}\) (C) \(\frac{2}{3}\) (D) \(\frac{2 \sqrt{2}}{3}\)

If \(C\) is the mid point of \(A B\) and \(P\) is any point outside \(A B\), then (A) \(\overrightarrow{P A}+\overrightarrow{P B}=2 \overrightarrow{P C}\) (B) \(\overrightarrow{P A}+\overrightarrow{P B}=\overrightarrow{P C}\) (C) \(\overrightarrow{P A}+\overrightarrow{P B}+2 \overrightarrow{P C}=0\) (D) \(\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C}=0\)

The distance between the line \(\vec{r}=2 \hat{i}-2 \hat{j}+3 \hat{k}\) \(+\lambda(\hat{i}+\hat{j}+4 \hat{k})\) and the plane \(\vec{r} \cdot(\hat{i}+5 \hat{j}+\hat{k})=5\) is \([2005]\) (A) \(\frac{10}{9}\) (B) \(\frac{10}{3 \sqrt{3}}\) (C) \(\frac{3}{10}\) (D) \(\frac{10}{3}\)

For 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) \(3 \vec{a}^{2}\) (B) \(\vec{a}^{2}\) (C) \(2 \vec{a}^{2}\) (D) \(4 \vec{a}^{2}\)

If non-zero numbers \(a, b, c\) are in H.P., then the straight line \(\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0\) always passes through a fixed point. That point is \([2005]\) (A) \((-1,2)\) (B) \((-1,-2)\) (C) \((1,-2)\) (D) \(\left(1,-\frac{1}{2}\right)\)

Let \(a, b\) and \(c\) be distinct non-negative numbers. If the vectors \(a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}\) and \(c \hat{i}+c \hat{j}+b \hat{k}\) lie in a plane, then \(c\) is \([2005]\) (A) the Geometric Mean of \(a\) and \(b\) (B) the Arithmetic Mean of \(a\) and \(b\) (C) equal to zero (D) the Harmonic Mean of \(a\) and \(b\)

Let \(\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k} \quad\) and \(\vec{c}=y \hat{i}+x \hat{j}\) \(+(1+x-y) \hat{k} .\) Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on \(\quad[\mathbf{2 0 0 5}]\) (A) only \(y\) (B) only \(x\) (C) both \(x\) and \(y\) (D) neither \(x\) nor \(y\)

The values of \(\mathrm{a}\), for which the points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) with position vectors \(2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}\) and \(a \hat{i}-3 \hat{j}+\hat{k}\) respectively are the vertices of a right-angled triangle with \(C=\frac{\pi}{2}\) are \([\mathbf{2 0 0 6}]\) (A) 2 and 1 (B) \(-2\) and \(-1\) (C) \(-2\) and 1 (D) 2 and \(-1\)

If \(\hat{u}\) and \(\hat{v}\) are unit vectors and \(\theta\) is the acute angle between them, then \(2 \hat{u} \times 3 \hat{v}\) is a unit vector for \([\mathbf{2 0 0 7}]\) (A) exactly two values of \(\theta\) (B) more than two values of \(\theta\) (C) no value of \(\bar{\theta}\) (D) exactly one value of \(\theta\)

Let \(\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}-\hat{j}+2 \hat{k}\) and \(\bar{c}=x \hat{i}+(x-2) \hat{j}-\hat{k} .\) If the vector \(\bar{c}\) lies in the plane of \(\bar{a}\) and \(\bar{b}\), then \(x\) equals [2007] (A) 0 (B) 1 (C) \(-4\) (D) \(-2\)

The vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}\) lies in the plane of the vectors \(\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\hat{j}+\hat{k}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then which one of the following gives possible values of \(\alpha\) and \(\beta ?\) (A) \(\alpha=2, \beta=2\) (B) \(\alpha=1, \beta=2\) (C) \(\alpha=2, \beta=1\) (D) \(\alpha=1, \beta=1\)

The non-zero vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) are related by \(\vec{a}=8 \vec{b}\) and \(\vec{c}=7 \vec{b}\). Then the angle between \(\vec{a}\) and \(\vec{c}\) is \([\mathbf{2 0 0 8}]\) (A) 0 (B) \(\pi / 4\) (C) \(\pi / 2\) (D) \(\pi\)

If \(\vec{u}, \vec{v}, \vec{w}\) are non-coplanar vectors and \(p, q\) are real numbers, then the equality \([3 \vec{u} p \vec{v} \quad p \vec{w}]-[p \vec{v} \vec{w} q \vec{u}]\) \(-\left[\begin{array}{lll}2 \vec{w} & q \vec{v} & q \vec{u}\end{array}\right]=0\) holds for [2009] (A) exactly one value of \((p, q)\) (B) exactly two values of \((p, q)\) (C) more than two but not all values of \((p, q)\) (D) all values of \((p, q)\)

The projections of a vector on the three coordinate axis are \(6,-3,2\) respectively. The direction cosines of the vector are (A) \(6,-3,2\) (B) \(\frac{6}{5},-\frac{3}{5}, \frac{2}{5}\) (C) \(\frac{6}{7},-\frac{3}{7}, \frac{2}{7}\) (D) \(-\frac{6}{7},-\frac{3}{7}, \frac{2}{7}\)

Let \(\vec{a}=\vec{j}-\vec{k}\) and \(\vec{c}=\vec{i}-\vec{j}-\vec{k}\). Then, the vector \(\vec{b}\) satisfying \(\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0}\) and \(a \cdot \vec{b}=3\) is (A) \(2 \hat{i}-\hat{j}+2 \hat{k}\) (B) \(\hat{i}-\hat{j}-2 \hat{k}\) (C) \(\hat{i}+\hat{j}-2 \hat{k}\) (D) \(-\hat{i}+\hat{j}-2 \hat{k}\)

If the vectors \(\vec{a}=\hat{i}-\hat{j}+2 \hat{k}, \hat{b}=2 \hat{i}+4 \hat{j}+\hat{k}\) and \(\vec{c}=\lambda \hat{i}+\hat{j}\) \(+\mu \hat{k}\) are mutually orthogonal, then the tuple \((\lambda, \mu)=\) \([\mathbf{2 0 1 0}]\) (A) \((2,-3)\) (B) \((-2,3)\) (C) \((3,-2)\) (D) \((-3,2)\)

The vectors \(\vec{a}\) and \(\vec{b}\) are not perpendicular and \(\vec{c}\) and \(\vec{d}\) are two vectors satisfying: \(\vec{b} \times \vec{c}=\vec{b} \times \vec{d}\) and \(\vec{a} \cdot \vec{d}=0 .\) Then, the vector \(\vec{d}\) is equal to (A) \(c+\left(\frac{a \cdot c}{a \cdot b}\right) b\) (B) \(b+\left(\frac{b . c}{a \cdot b}\right) c\) (C) \(c-\left(\frac{a . c}{a \cdot b}\right) b\) (D) \(b-\left(\frac{b . c}{a \cdot b}\right) c\)

Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors. If the vectors \(\vec{c}=\hat{a}+2 \hat{b}\) and \(d=5 \hat{a}-4 \hat{b}\) are perpendicular to each other, then the angle between \(\hat{a}\) and \(\hat{b}\) is (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{\pi}{4}\)

Let \(A B C D\) be a parallelogram such that \(\overrightarrow{A B}=\vec{q}, \overrightarrow{A D}=\bar{p}\) and \(\square B A D\) be an acute angle. If \(\vec{r}\) is the vector which coincides with the altitude directed from the vertex \(B\) to the side \(A D\), then \(\vec{r}\) is given by \([2012]\) (A) \(\vec{r}=3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (B) \(\vec{r}=-\vec{q}+\left(\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}\right) \vec{p}\)

If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}+2 \hat{j}+4 \hat{k}\) represent the sides of a triangle \(A B C\), then the length of the median through \(A\) is [2013] (A) \(\sqrt{72}\) (B) \(\sqrt{33}\) (C) \(\sqrt{45}\) (D) \(\sqrt{18}\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that no two of them are collinear and \((\vec{a} \times \vec{b}) \times \vec{c}=\frac{1}{3}|\vec{b}||\vec{c}| \vec{a}\). If \(\theta\) is the angle between vectors \(\vec{b}\) and \(\vec{c}\), then a value of \(\sin \theta\) (A) \(\frac{-\sqrt{2}}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{-2 \sqrt{3}}{3}\) (D) \(\frac{2 \sqrt{2}}{3}\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\sqrt{3}}{2}(\vec{b}+\vec{c})\). if \(\vec{b}\) is not parallel to \(\vec{c}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is (A) \(\frac{5 \pi}{6}\) (B) \(\frac{3 \pi}{4}\) (C) \(\frac{\pi}{2}\) (D) \(\frac{2 \pi}{3}\)