Chapter 21

\(a\) and \(b\) are mutually perpendicular unit vectors. If \(r\) is a vector satisfying \(r \cdot a=0, r \cdot b=1\) and \([r a b]=1\), then \(r\) is (A) \(a \times b+b\) (B) \(a+(a \times b)\) (C) \(b+(a \times b)\) (D) \(a \times \vec{b}+a\)

\(a, b, c\) are three vectors of magnitude, \(\sqrt{3}, 1,2\) such that \(a \times(a \times c)+3 b=O .\) If \(\theta\) is the angle between \(a\) and \(c\), then \(\cos ^{2} \theta\) is equal to (A) \(\frac{1}{4}\) (B) \(\frac{3}{4}\) (C) \(\underline{1}=\) (D) none of these

Let \(A B C D E F\) be a regular hexagon. If \(A D=x B C\) and \(C F=y A B\), then \(x y=\) (A) 4 (B) \(-4\) (C) 2 (D) \(-2\)

Given a cube \(A B C D A_{1} B_{1} C_{1} D_{1}\) with lower base \(A B C D\), upper base \(A_{1} B_{1} C_{1} D_{1}\) and the lateral edges \(A A_{1}, B B_{1}\), \(C C_{1}\) and \(D D_{1} ; M\) and \(M_{1}\) are the centres of the faces \(A B C D\) and \(A_{1} B_{1} C_{1} D_{1}\) respectively. \(O\) is a point on line \(M M_{1}\), such that \(O A+O B+O C+O D=O M_{1}\), then \(O M=\lambda O M_{1}\) if \(\lambda=\) (A) \(\frac{1}{4}\) (B) \(\frac{1}{2}\) (C) \(\frac{1}{6}\) (D) \(\frac{1}{8}\)

The triangle \(A B C\) is defined by the vertices \(A(1,-2,2)\), \(B(1,4,0)\) and \(C(-4,1,1) .\) Let \(M\) be the foot of the altitude drawn from the vertex \(B\) to side \(A C\). Then, \(B M=\) (A) \((-20 / 7,-30 / 7,10 / 7)\) (B) \((-20,-30,10)\) (C) \((2,3,-1)\) (D) none of these

If \(A B=3 i+j-k\) and \(A C=i-j+3 k .\) If the point \(P\) on the line segment \(B C\) is equidistant from \(A B\) and \(A C\), then \(A P\) is (A) \(2 i-k\) (B) \(i-2 \mathrm{k}\) (C) \(2 i+k\) (D) none of these

\(A, B, C, D\) are four points on a plane with position vectors \(a, b, c, d\), respectively, such that \((a-d)\). \((b-c)=(b-d) \cdot(c-a)=0 .\) For \(\Delta A B C, D\) is the (A) incentre (B) orthocentre (C) centroid (D) none of these

If \(a\) and \(b\) are two unit vectors, then the vector \((a+b)\) \(\times(a \times b)\) is parallel to the vector (A) \(a-b\) (B) \(a+b\) (C) \(2 a-b\) (D) \(2 a+b\)

In a parallelogram \(A B C D,|A B|=a,|A D|=b\) and \(\mid A C\) \(\mid=c\). Then, \(D B \cdot A B\) has the value (A) \(\frac{3 a^{2}+b^{2}-c^{2}}{2}\) (B) \(\frac{a^{2}+3 b^{2}-c^{2}}{2}\) (C) \(\frac{a^{2}-b^{2}+3 c^{2}}{2}\) (D) \(\frac{a^{2}+3 b^{2}+c^{2}}{2}\)

If \(a, b, c\) are non-coplanar unit vectors such that \(a \times(b \times c)=\frac{b+c}{\sqrt{2}}\), then the angle between \(a\) and \(b\) is (A) \(\frac{3 \pi}{4}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{2}\) (D) \(\pi\)

If the vectors \(a\) and \(b\) are perpendicular to each other, then a vector \(v\) in terms of \(a\) and \(b\) satisfying the equations \(v \cdot a=0, v \cdot b=1\) and \([v a b]=1\) is (A) \(\frac{1}{|b|^{2}} b+\frac{1}{|a \times b|^{2}} a \times b\) (B) \(\frac{b}{|b|}+\frac{a \times b}{|a \times b|^{2}}\) (C) \(\frac{b}{|b|^{2}}+\frac{a \times b}{|a \times b|}\) (D) none of these

Let the unit vectors \(a\) and \(b\) be perpendicular to each other and the unit vector \(c\) be inclined at an angle \(\theta\) to both \(a\) and \(b\). If \(c=x a+y b+z(a \times b)\), then (A) \(x=\cos \theta, y=\sin \theta, z=\cos 2 \theta\) (B) \(x=\sin \theta, y=\cos \theta, z=-\cos 2 \theta\) (C) \(x=y=\cos \theta, z^{2}=\cos 2 \theta\) (D) \(x=y=\cos \theta, z^{2}=-\cos 2 \theta\)

If \(\sum_{i=1}^{n} a_{i}=0\) where \(\left|a_{i}\right|=1+i\), then the value of is \(\sum_{1 \leq i

Forces \(P, Q\) act at \(O\) and have a resultant \(R\). If any transversal cuts their lines of action at \(A, B, C\), respectively, then (A) \(\frac{P}{O A}+\frac{Q}{O B}+\frac{R}{O C}=0\) (B) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}+\frac{\mathrm{R}}{O C}=1\) (C) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}-\frac{\mathrm{R}}{O C}=0\) (D) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}-\frac{\mathrm{R}}{O C}=1\).

A vector \(A\) has components \(A_{1}, A_{2}, A_{3}\) in a right-handed rectangular cartesian coordinate system \(O x, O y, O z\). The coordinate system is rotated about the z-axis through an angle \(\frac{\pi}{2} .\) The components of \(A\) in the new coordinate system are (A) \(A_{1},-A_{2}, A_{3}\) (B) \(A_{2}, A_{1}, A_{3}\) (C) \(A_{1}, A_{2},-A_{3}\) (D) \(A_{2},-A_{1}, A_{3}\).

In a \(\Delta O A B, \mathrm{E}\) is the mid-point of \(O B\) and \(D\) is a poir on \(A B\) such that \(A D: D B=2: 1 .\) If \(O D\) and \(A E\) intes sect at \(P\), then the ratio \(O P: P D\) is (A) \(1: 2\) (B) \(2: 1\) (C) \(3: 2\) (D) \(2: 3\).

If \(a, b, c\) are three non-parallel unit vectors such that \(a \times(b \times c)=\frac{1}{2} b\), then the angles which a makes with \(b\) and \(c\) are (A) \(90^{\circ}, 60^{\circ}\) (B) \(45^{\circ}, 60^{\circ}\) (C) \(30^{\circ}, 60^{\circ}\) (D) none of these

If the \(p\) th, \(q\) th and \(r\) th terms of a G. P. are positive numbers \(a, b\) and \(c\), respectively, then the angle between the vectors \(i l_{n} a+j l_{n} b+k l_{n} c\) and \(i(q-r)+j(r-p)+k\) \((p-q)\) is (A) \(\frac{\pi}{3}\) (B) \(\frac{\pi}{6}\) (C) \(\frac{\pi}{2}\) (D) none of these

A vector \(a\) is collinear with vector \(b=\left(6,-8,-7 \frac{1}{2}\right)\) and make an acute angle with the positive direction of \(z\)-axis. If \(|a|=50\), then \(a=\) (A) \((24,32,30)\) (B) \((24,-32,30)\) (C) \((-24,32,30)\) (D) none of these

The perpendicular distance of a corner of a unit cube form a diagonal not passing through it is (A) \(\sqrt{6}\) (B) \(\frac{\sqrt{6}}{3}\) (C) \(\frac{3}{\sqrt{6}}\) (D) none of these

The vectors \(a, b\) and \(c\) are equal in length and taken pairwise, they make equal angles. If \(a=i+j, b=j+\) \(k\), and \(c\) makes an obtuse angle with the base vector \(i\), then \(c\) is equal to (A) \(i+k\) (B) \(-i+4 j-k\) (C) \(\frac{-1}{3} i+\frac{4}{3} j-\frac{1}{3} k\) (D) \(\frac{1}{3} i+\frac{-4}{3} j+\frac{1}{3} k\).

If the four points \(a, b, c, d\) are coplanar, then \(\left[\begin{array}{ll}b c d\end{array}\right]+\left[\begin{array}{ll}c & a & d\end{array}\right]+\left[\begin{array}{ll}a & b & d\end{array}\right]=\) (A) 0 (B) 1 (C) \(-1\) (D) \(\left[\begin{array}{lll}a & b & c\end{array}\right]\)

A tetrahedron has vertices at \(O(0,0,0), A(1,2,1)\), \(B(2,1,3)\) and \(C(-1,1,2)\). Then, the angle between the faces \(O A B\) and \(A B C\) will be (A) \(\cos ^{-1}\left(\frac{19}{35}\right)\) (B) \(\cos ^{-1}\left(\frac{71}{31}\right)\) (C) \(30^{\circ}\) (D) \(90^{\circ}\)

If a quadrilateral \(A B C D\) is such that \(A B=b, A D=d\) and \(A C=m b+p d(m+p \geq 1)\), then the area of the quadrilateral is \(k(p+m)|b \times d|\), where \(k\) is equal to (A) \(\frac{1}{4}\) (B) \(\frac{1}{8}\) (C) \(\frac{1}{2}\) (D) none of these

Let \(a\) be a unit vector and \(b\) be a non-zero vector not parallel to \(a\). If two sides of the triangle are represented by the vectors \(\sqrt{3}(a \times b)\) and \(b-(a \cdot b) a\), then the angles of the triangle are (A) \(30^{\circ}, 90^{\circ}, 60^{\circ}\) (B) \(45^{\circ}, 45^{\circ}, 90^{\circ}\) (C) \(60^{\circ}, 60^{\circ}, 60^{\circ}\) (D) none of these

Let \(u\) and \(v\) be unit vectors. If \(w\) is a vector such that \(w\) \(+(w \times u)=v\), then \(|(u \times v) \cdot w|\) \((\mathrm{A}) \leq \frac{1}{3}\) \((\mathrm{B}) \leq \frac{1}{2}\) (C) \(>\frac{1}{3}\) \((\mathrm{D}) \geq \frac{1}{2}\)

If \(b\) and \(c\) are any two non-collinear unit vectors and \(a\) is any vector, then \((a \cdot b) b+(a \cdot c) c+\frac{a \cdot(b+c)}{|b+c|^{2}}(b \times c)=\) (A) \(a\) (B) \(b\) (C) \(c\) (D) none of these

If the vector \(-i+j-k\) bisects the angle between \(3 i+4 j\) and vector \(c\), then the unit vector along \(c\) is (A) \(\frac{-11 i-10 j-2 k}{15}\) (B) \(\frac{-11 i+10 j+2 k}{15}\) (C) \(\frac{-11 i+10 j-2 k}{15}\) (D) none of these

If \(a, b\) and \(c\) are three unit vectors such that \(a+b+c\) is also a unit vector and \(\theta_{1}, \theta_{2}\) and \(\theta_{3}\) are angles between the vectors \(a, b ; b, c\) and \(c, a\), respectively, then among \(\theta_{1}, \theta_{2}\) and \(\theta_{3}\) (A) all are acute angles (B) all are right angles (C) at least one is obtuse angle (D) none of these

If \(x\) and \(y\) are two non-collinear vectors and \(A B C\) is a triangle with side lengths \(a, b, c\) satisfying \((20 a-15 b) x+(15 b-12 c) y+(12 c-20 a)(x \times y)=\overrightarrow{0}\) then \(\triangle A B C\) is (A) an acute-angled triangle (B) an obtuse-angled triangle (C) a right-angled triangle (D) an isosceles triangle

Let \(a=i+j\) and \(b=2 i-k\). The point of intersection of the lines \(r \times a=b \times a\) and \(r \times b=a \times b\) is (A) \(-i+j+k\) (B) \(3 i-j+k\) (C) \(3 i+j-k\) (D) \(i-j-k\)

The sides of a parallelogram are \(2 i+4 j-5 k\) and \(i+2 j\) \(+3 k\). The unit vector parallel to one of the diagonals size is (A) \(\frac{1}{7}(3 i+6 j-2 k)\) (B) \(\frac{1}{7}(3 i-6 j-2 k)\) (C) \(\frac{1}{7}(-3 i+6 j-2 k)\) (D) \(\frac{1}{7}(3 i+6 j+2 k)\)

If \(a \times(b \times c)+(a \cdot b) b=(4-2 \beta-\sin \alpha) b+\left(\beta^{2}-1\right) c\) and \((c \cdot c) a=c\), while \(b\) and \(c\) are non-collinear, then (A) \(\alpha=\frac{\pi}{2}\) (B) \(\alpha=\frac{\pi}{3}\) (C) \(\beta=1\) (D) \(\beta=-1\)

Let \(b=4 i+3 j\) and \(c\) be two vectors perpendicular to each other in \(x y\)-plane, then the vector in the same plane having projections 1 and 2 along \(b\) and \(c\) respectively is (A) \(2 i-j\) (B) \(-2 i+j\) (C) \(2 i+j\) (D) none of these

A vector of magnitude 2 along a bisector of the angle between the two vectors \(2 i-2 j+k\) and \(i+2 j-2 k\) is (A) \(\frac{2}{\sqrt{10}}(3 i-k)\) (B) \(\frac{1}{\sqrt{26}}(i-4 j+3 k)\) (C) \(\frac{2}{\sqrt{26}}(i-4 j+3 k)\) (D) none of these

The value of \(\lambda\) such that \((x, y, z) \neq(0,0,0)\) and \((i+j+3 k) x+(3 i-3 j+k) y+(-4 i+5 j) z\) \(=\lambda(x i+y j+z k)\) is (A) 0 (B) 1 (C) \(-1\) (D) none of these

If the three vectors \(a=(12,4,3), b=(8,-12,-9)\) and \(c=(33,-4,-24)\) define a parallelopiped, then (A) the lengths of the edges are \(13,17,41\) (B) areas of the faces are \(220,435,455\) (C) volume of parallelopiped is 3696 (D) all of these

A vector of magnitude \(\sqrt{51}\) which makes equal angles with the vectors \(a=\frac{1}{3}(i-2 j+2 k)\), \(b=\frac{1}{5}(-4 i-3 k)\) and \(c=j\) is given by (A) \(5 i-j-5 k\) (B) \(-5 i+j+5 k\) (C) \(5 i+j+5 k\) (D) none of these

If \(D A=a, A B=b\) and \(C B=k a\), where \(k>0\) and \(X, Y\) are the mid-points of \(D B\) and \(A C\) respectively such that \(|a|=17\) and \(|X Y|=4\), then \(k\) is equal to (A) \(\frac{8}{17}\) (B) \(\frac{9}{17}\) (C) \(\frac{25}{17}\) (D) 1

Let \(a\) and \(b\) be two non-collinear unit vectors. If \(u=a-(a \cdot b) b\) and \(v=a \times b\), then \(|v|\) is (A) \(|u|\) (B) \(|u|+|u \cdot a|\) (C) \(|u|+|u \cdot b|\) (D) \(|u|+u \cdot(a+b)\)

A non-zero vector \(a\) is parallel to the line of intersection of the plane determined by the vectors \(i, i+j\) and the plane determined by the vectors \(i-j, i+k\). The angle between \(a\) and the vector \(i-2 j+2 k\) is (A) \(\frac{\pi}{3}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{3 \pi}{4}\) (D) none of these

A unit vector coplanar with \(i+j+2 k\) and \(i+2 j+k\) and perpendicular to \(i+j+k\) is (A) \(\frac{j-k}{\sqrt{2}}\) (B) \(\frac{-j+k}{\sqrt{2}}\) (C) \(\frac{j+k}{\sqrt{2}}\) (D) \(\frac{-(j+k)}{\sqrt{2}}\)

The vectors \(a, b, c\) are of same length and taken pairwise, they form equal angles. If \(a=i+j\) and \(b=j+k\), then \(c=\) (A) \(i+k\) (B) \(j+k\) (C) \(i+k\) (D) \(-\frac{i}{3}+\frac{4}{3} j-\frac{k}{3}\)

\(a\) and \(c\) are unit vectors and \(|b|=4\) with \(a \times b=2 a \times c\). The angle between \(a\) and \(c\) is \(\cos ^{-1}\left(\frac{1}{4}\right)\). Then, \(b-2 c\) \(=\mathrm{la}\), if \(\lambda\) is (A) 3 (B) \(-3\) (C) 4 (D) \(-4\)

The vector c directed along the bisectors of the angle between the vectors \(a=7 i-4 j-4 k\) and \(\hat{b}=-2 i-j+2 k\) if \(|c|=3 \sqrt{6}\), is given by (A) \(i-7 j+2 k\) (B) \(2 i+7 j-3 k\) (C) \(-i+7 j-2 k\) (D) \(4 i+7 j-4 k\)

The vector differential operator DEL, written \(\nabla\), is defined by \(\nabla=\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k=i \frac{\partial}{\partial x}+j \frac{\partial}{\partial y}+k \frac{\partial}{\partial z}\), where \(\frac{\partial}{\partial x}\) rep- resents the derivative w.r.t. \(x\) regarding \(y\) and \(z\) as constant. Similarly, \(\frac{\partial}{\partial y}\) represents the derivative w.r.t. \(y\) regarding \(x\) and \(z\) as constant and \(\frac{\partial}{\partial z}\) represents the derivative w.r.t. \(z\) regarding \(x\) and \(y\) as constant. The operator \(\nabla\) is also known as nabla. Let \(\phi(x, y, z)\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the gradient of \(\phi\), written \(\nabla \phi\) or grad \(\phi\), is defined by $$ \nabla \phi=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \phi=\frac{\partial \phi}{\partial x} i+\frac{\partial \phi}{\partial y} j+\frac{\partial \phi}{\partial z} k $$ Let \(r\) be any vector such that \(r=x i+y j+z k\) If \(\phi=\ln |r|\) then \(\nabla \phi=\) (A) \(\frac{r}{r^{2}}\) (B) \(\frac{r}{r^{3}}\) (C) \(\frac{r}{r^{4}}\) (D) \(\frac{r}{r}\)

The vector differential operator DEL, written \(\nabla\), is defined by \(\nabla=\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k=i \frac{\partial}{\partial x}+j \frac{\partial}{\partial y}+k \frac{\partial}{\partial z}\), where \(\frac{\partial}{\partial x}\) rep- resents the derivative w.r.t. \(x\) regarding \(y\) and \(z\) as constant. Similarly, \(\frac{\partial}{\partial y}\) represents the derivative w.r.t. \(y\) regarding \(x\) and \(z\) as constant and \(\frac{\partial}{\partial z}\) represents the derivative w.r.t. \(z\) regarding \(x\) and \(y\) as constant. The operator \(\nabla\) is also known as nabla. Let \(\phi(x, y, z)\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the gradient of \(\phi\), written \(\nabla \phi\) or grad \(\phi\), is defined by $$ \nabla \phi=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \phi=\frac{\partial \phi}{\partial x} i+\frac{\partial \phi}{\partial y} j+\frac{\partial \phi}{\partial z} k $$ Let \(r\) be any vector such that \(r=x i+y j+z k\) \(\nabla r^{n}=\) (A) \(n r n-{ }^{1} r\) (B) \(n m-{ }^{2} r\) (C) \(\bar{n} \overline{r n} r\) (D) none of these

Let \(\nabla(x, y, z)=V_{1} i+V_{2} j+V_{3} k\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the divergence of \(V\), written \(\nabla \cdot V\) or div \(V\) is defined by $$ \begin{aligned} \nabla \cdot V &=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \cdot\left(V_{1} i+V_{2} j+V_{3} k\right) \\ &=\frac{\partial V_{1}}{\partial x}+\frac{\partial V_{2}}{\partial y}+\frac{\partial V_{3}}{\partial z} \end{aligned} $$ Here, \(\nabla=\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\) is the del operator. Note the analogy with \(A \cdot B=A_{1} B_{1}+A_{2} B_{2}+A_{3} B_{3} .\) Also, note that \(\vec{\nabla} \cdot V \neq V \cdot \nabla\). If \(\phi=2 x^{3} y^{2} z^{4}\), then \(\nabla . \nabla \phi=k\left(3 x y^{2} z^{4}+x^{3} z^{4}+6 x^{3} y^{2} z^{2}\right)\), where \(k=\) (A) 2 (B) 3 (C) 4 (D) 6

Let \(\nabla(x, y, z)=V_{1} i+V_{2} j+V_{3} k\) be defined and differentiable at each point \((x, y, z)\) in a certain region of space. Then, the divergence of \(V\), written \(\nabla \cdot V\) or div \(V\) is defined by $$ \begin{aligned} \nabla \cdot V &=\left(\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\right) \cdot\left(V_{1} i+V_{2} j+V_{3} k\right) \\ &=\frac{\partial V_{1}}{\partial x}+\frac{\partial V_{2}}{\partial y}+\frac{\partial V_{3}}{\partial z} \end{aligned} $$ Here, \(\nabla=\frac{\partial}{\partial x} i+\frac{\partial}{\partial y} j+\frac{\partial}{\partial z} k\) is the del operator. Note the analogy with \(A \cdot B=A_{1} B_{1}+A_{2} B_{2}+A_{3} B_{3} .\) Also, note that \(\vec{\nabla} \cdot V \neq V \cdot \nabla\). If \(r=x i+y j+z k\), then \(\nabla \cdot\left(\frac{r}{r^{3}}\right)=\) (A) 0 (B) 1 (C) \(-1\) (D) none of these

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 scalar function \(\phi\), possessing continuous secot order partial derivatives \(\nabla \times(\nabla \phi)=\) (A) \(\phi\) (B) 0 (C) \(\nabla \phi\) (D) none of these