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) \(a \times \bar{b}+a\)

\(a, b, c\) 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) \(\frac{1}{2}\) (D) none of these

A vector \(a\) has components \(2 p\) and 1 w.r.t. a rectangular cartesian system. This system is rotated through acertain angle about the origin in the counter- clockwise sense. If w.r.t. the new system, \(a\) has components \(p+1\) and 1 , then (A) \(p=0\) (B) \(p=1\) or \(p=-\frac{1}{3}\) (C) \(p=-1\) or \(p=\frac{1}{3}\) (D) \(p=1\) or \(p=-1\)

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) \(-2\)

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) \(-2\)

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) \((-20 / 7,-307,10 / 7)\) (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

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) \(2 a+b\)

If \(r=\lambda(a \times b)+\mu(b \times c)+v(c \times a)\) and \([a b c]=\frac{1}{8}\), then \(\lambda+\mu+v\) is equal to (A) \(r \cdot(a+b+c)\) (B) \(8 r-(a+b+c)\) (C) \(4 r \cdot(a+b+c)\) (D) none of these

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) \(\frac{a^{2}+3 b^{2}+c^{2}}{2}\)

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 \(a, c, d\) are non-coplanar vectors and \(d \cdot\\{a \times[b \times\) \((c \times d)]\\}\) is equal to (A) \((b \cdot d)[a c d]\) (B) \((a \cdot d)[a c d]\) (C) \((c \cdot d)[a c d]\) (D) none of these

If \(4 a+5 b+9 c=0\), then \((a \times b) \times[(b \times c) \times(c \times a)]\) is equal to (A) A vector perpendicular to the plane of \(a, b\) and \(c\) (B) A scalar quantity (C) 0 (D) none of these

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) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}+\frac{\mathrm{R}}{O C}=0\) (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) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}-\frac{\mathrm{R}}{O C}=1\).

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) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}+\frac{\mathrm{R}}{O C}=0\) (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) \(\frac{\mathrm{P}}{O A}+\frac{\mathrm{Q}}{O B}-\frac{\mathrm{R}}{O C}=1\).

In a \(\Delta O A B, \mathrm{E}\) is the mid-point of \(O B\) and \(D\) is a point on \(A B\) such that \(A D: D B=2: 1 .\) If \(O D\) and \(A E\) intersect 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

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 \([b c d]+[c a d]+[a b d]=\) (A) 0 (B) 1 (C) \(-1\) (D) \([a b c]\)

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) \(90^{\circ}\)

Let \(a\) be a unit vector and \(b\) be a non-zero vector not parallel to \(a\). If two sides of the triangle are representedby 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 \(|(u \times v) \cdot w|\) \((\mathrm{A}) \leq \frac{1}{3}\) (B) \(\leq \frac{1}{2}\) (C) \(>\frac{1}{3}\) (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) \(\bar{b}\) (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 \(x+y=a, x \times y=b\) and \(x \cdot a=1\), then (A) \(x=\frac{a+a \times b}{a^{2}}\) (B) \(y=\frac{\left(a^{2}-1\right) a-a \times b}{a^{2}}\) (C) \(x=\frac{b+a \times b}{a^{2}}\) (D) \(y=\frac{\left(b^{2}-1\right) b-a \times b}{a^{2}}\)

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

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

A unit vector \(a\) makes an angle \(\frac{\pi}{4}\) with \(i\) and \(\frac{\pi}{3}\) with \(j\). If the angle between \(a\) and \(k\) is \(\theta\), where \(\theta\) is acute, then (A) \(a=\frac{1}{2} i+\frac{1}{\sqrt{2}} j+\frac{1}{2} k\) (B) \(a=\frac{1}{\sqrt{2}} i+\frac{1}{2} j+\frac{1}{2} k\) (C) \(\theta=\frac{\pi}{3}\) (D) \(\theta=\frac{\pi}{6}\)

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

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

The position vectors of two points \(A\) and \(C\) are \(9 i-j+\) \(7 k\) and \(7 i-2 j+7 k\), respectively. The point of intersection of vectors \(A B=4 i-j+3 k\) and \(C D=2 i-j+2 k\) is \(P\). If vector \(P Q\) is perpendicular to \(A B\) and \(C D\) and \(P Q\) \(=15\) units, the position vector of \(Q\) is (A) \(6 i-9 j-9 k\) (B) \(-4 i+11 j+11 k\) (C) \(6 i+9 j-9 k\) (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, \(b-2 c\) \(=1 a\), if \(\lambda\) is(A) 3 (B) \(-3\) (C) 4 (D) \(-4\)

The vector c directed along the bisectors of the angl 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) \(4 i+7 j-4 k\)

If \(\phi=\frac{1}{r}\), then \(\nabla \phi=\) (A) \(\frac{r}{r}\) (B) \(\frac{r}{r^{2}}\) (C) \(\frac{r}{r^{3}}\) (D) \(-\frac{r}{r^{3}}\)

If \(r=x i+y j+z k\) then \(\nabla^{2}\left(\frac{1}{r}\right)=\) (A) 1 (B) \(-1\) (C) 0 (D) none of these

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