Chapter 25

Let \(a, b, c \in \mathbf{R}\) be such that \(a^{2}+b^{2}+c^{2}=1\). If \(a \cos \theta-b \cos \left(\theta+\frac{2 \pi}{3}\right)=c \cos \left(\theta+\frac{4 \pi}{3}\right)\), where \(\theta=\frac{\pi}{9}\) then the angle between the vectors \(a \hat{i}+b \hat{j}+c \hat{k}\) and \(b \hat{i}+c \hat{j}+a \hat{k}\) is: \([\) Sep. \(03,2020(\) II \()]\) (a) \(\frac{\pi}{2}\) (b) \(\frac{2 \pi}{3}\) (c) \(\frac{\pi}{9}\) (d) 0

Let the position vectors of points ' \(A\) ' and ' \(B^{\prime}\) be \(\hat{i}+\hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{j}+3 \hat{k}\), respectively. A point ' \(\mathrm{P}\) ' divides the line segment \(\mathrm{AB}\) internally in the ratio \(\lambda: 1(\lambda>0)\). If \(O\) is the region and \(\overrightarrow{O B} \cdot \overrightarrow{O P}-3|\overrightarrow{O A} \times \overrightarrow{O P}|^{2}=6\), then \(\lambda\) is equal to

If the vectors, \(\bar{p}=(a+1) \hat{i}+a \hat{j}+a \hat{k}, \vec{q}=a \hat{i}+(a+1) \hat{j}+a \hat{k}\) and \(\vec{r}=a \hat{i}+a \hat{j}+(a+1) \hat{k} \quad(a \in \mathrm{R})\) are coplanar and \(3(\vec{p} \cdot \vec{q})^{2}-\lambda|\vec{r} \times \vec{q}|^{2}=0\), then the value of \(\lambda\) is

Let \(\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}\) and \(\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}\) be two vectors. If a vector perpendicular to both the vectors \(\vec{a}+\vec{b}\) and \(\vec{a}-\vec{b}\) has the magnitude 12 then one such vector is : [April 12, 2019 (I)] (a) \(4(2 \hat{i}+2 \hat{j}+2 \hat{k})\) (b) \(4(2 \hat{i}-2 \hat{j}-\hat{k})\) (c) \(4(2 \hat{i}+2 \hat{j}-\hat{k})\) (d) \(4(-2 \hat{i}-2 \hat{j}+\hat{k})\)

If the volume of parallelopiped formed by the vectors \(\hat{i}+\lambda \hat{j}+\hat{k}, \hat{j}+\lambda \hat{k}\) and \(\lambda \hat{i}+\hat{k}\) is minimum, then \(\lambda\) is equal to: \(\quad\) [April 12, 2019 (I)] (a) \(-\frac{1}{\sqrt{3}}\) (b) \(\frac{1}{\sqrt{3}}\) (c) \(\sqrt{3}\) (d) \(-\sqrt{3}\)

Let \(\alpha \in \mathrm{R}\) and the three vectors \(\vec{a}=\alpha \hat{i}+\hat{j}+3 \hat{k}\), \(\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}\) and \(\vec{c}=\alpha \hat{i}-2 \hat{j}+3 \hat{k}\). Then the set \(S=(\alpha: \vec{a}, \vec{b}\) and \(\vec{c}\) are coplanar) \(\quad\) [April 12, 2019 (II)] (a) is singleton (b) is empty (c) contains exactly two positive numbers (d) contains exactly two numbers only one of which is positive

If a unit vector \(\vec{a}\) makes angles \(\pi / 3\) with \(\hat{i}, \pi / 4\) with \(\hat{j}\) and \(\theta \in(0, \pi)\) with \(\hat{k}\), then a value of , is: [April 09,2019 (II)] (a) \(\frac{5 \pi}{6}\) (b) \(\frac{\pi}{4}\) (c) \(\frac{5 \pi}{12}\) (d) \(\frac{2 \pi}{3}\)

The sum of the distinct real values of \(\mu\), for which the vectors, \(\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}\), are co-planar, is : (a) \(-1\) (b) 0 (c) 1 (d) 2

The sum of the distinct real values of \(\mu\), for which the vectors, \(\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}\), are co-planar, is : (a) \(-1\) (b) 0 (c) 1 (d) 2

Let \(\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}\) and \(\beta \hat{i}+(1-\beta) \hat{j}\) respectively be the position vectors of the points \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) with respect to the origin \(\mathrm{O}\). If the distance of \(\mathrm{C}\) from the bisector of the acute angle between \(\mathrm{OA}\) and \(\mathrm{OB}\) is \(\frac{3}{\sqrt{2}}\), then the sum of all possible values of \(\beta\) is : [Jan. 11, 2019 (II)] (a) 4 (b) 3 (c) 2 (d) 1

Let \(\vec{\alpha}=(\lambda-2) \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\) and \(\vec{\beta}=(4 \lambda-2) \overrightarrow{\mathrm{a}}+3 \overrightarrow{\mathrm{b}}\) be two given vectors where vectors \(\vec{a}\) and \(\vec{b}\) are non-collinear. The value of \(\lambda\) for which vectors \(\vec{\alpha}\) and \(\vec{\beta}\) are collinear, is: [Jan. 10, 2019 (II)] (a) \(-4\) (b) \(-3\) (c) 4 (d) 3

Let \(\mathrm{ABC}\) be a triangle whose circumcentre is at \(\mathrm{P}\). If the position vectors \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{P}\) are \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) and \(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{4}\) respectively, then the position vector of the orthocentre of this triangle, is: \(\quad\) Online April 10, 2016] (a) \(-\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{2}\right)\) (b) \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\) (c) \(\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{2}\right)\) (d) \(\overrightarrow{0}\)

If the vectors \(\overline{\mathrm{AB}}=3 \hat{i}+4 \hat{k}\) and \(\overline{\mathrm{AC}}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\), then the length of the median through \(\mathrm{A}\) is (a) \(\sqrt{18}\) (b) \(\sqrt{72}\) (c) \(\sqrt{33}\) (d) \(\sqrt{45}\)

If \(\vec{a}\) and \(\vec{b}\) are non-collinear vectors, then the value of \(\alpha\) for which the vectors \(\vec{u}=(\alpha-2) \vec{a}+\vec{b}\) and \(\vec{v}=(2+3 \alpha) \vec{a}-3 \vec{b}\) are collinear is: Online April 23, 2013] (a) \(\frac{3}{2}\) (b) \(\frac{2}{3}\) (c) \(-\frac{3}{2}\) (d) \(-\frac{2}{3}\)

If \(\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}-\hat{k}\) and \(\vec{c}=r \hat{i}+\hat{j}+(2 r-1) \hat{k}\) are three vectors such that \(\vec{c}\) is parallel to the plane of \(\vec{a}\) and \(\vec{b}\), then \(r\) is equal to [Online May 19, 2012] (a) 1 (b) \(-1\) (c) 0 (d) 2

Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-zero vectors which are pairwise non-collinear. If \(\vec{a}+3 \vec{b}\) is collinear with \(\vec{c}\) and \(\vec{b}+2 \vec{c}\) is collinear with \(\vec{a}\), then \(\vec{a}+3 \vec{b}+6 \vec{c}\) is : \(\quad\) [2011RS] (a) \(\vec{a}\) (b) \(\vec{c}\) (c) \(\overrightarrow{0}\) (d) \(\vec{a}+\vec{c}\)

If the \(p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+r \hat{k}(p \neq q \neq r \neq 1)\) vector are coplanar, then the value of \(p q r-(p+q+r)\) is [2011RS] (a) 2 (b) 0 (c) \(-1\) (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) \(\alpha=1, \beta=1\)

\(A B C\) is a triangle, right angled at \(A\). The resultant of the forces acting along \(\overline{A B}, \overline{B C}\) with magnitudes \(\frac{1}{A B}\) and \(\frac{1}{A C}\) respectively is the force along \(\overline{A D}\), where \(D\) is the foot of the perpendicular from \(A\) onto \(B C\). The magnitude of the resultant is (a) \(\frac{A B^{2}+A C^{2}}{(A B)^{2}(A C)^{2}}\) (b) \(\frac{(A B)(A C)}{A B+A C}\) (c) \(\frac{1}{A B}+\frac{1}{A C}\) (d) \(\frac{1}{A D}\)

If \(\mathrm{C}\) is the mid point of \(\mathrm{AB}\) and \(\mathrm{P}\) is any point outside \(\mathrm{AB}\), then \([\mathbf{2 0 0 5}]\) (a) \(\overrightarrow{P A}+\overrightarrow{P B}=2 \overrightarrow{P C}\) (b) \(\overrightarrow{P A}+\overrightarrow{P B}=\overrightarrow{P C}\) (c) \(\overline{P A}+\overrightarrow{P B}+2 \overrightarrow{P C}=\overrightarrow{0}\) (d) \(\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C}=\overrightarrow{0}\)

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 \(\hat{c} \hat{i}+c \hat{j}+b \hat{k}\) lie in a plane, then \(c\) is (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 \(b\)

If \(\vec{a}, \vec{b}, \vec{c}\) 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) no value of \(\lambda\) (b) all except one value of \(\lambda\) (c) all except two values of \(\lambda\) (d) all values of \(\lambda\)

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) 0 (b) \(\lambda \vec{b}\) (c) \(\lambda \vec{c}\) (d) \(\lambda \vec{a}\)

Consider points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) with position vectors \(7 \hat{i}-4 \hat{j}+7 \hat{k}, \hat{i}-6 \hat{j}+10 \hat{k},-\hat{i}-3 \hat{j}+4 \hat{k}\) and \(5 \hat{i}-\hat{j}+5 \hat{k}\) respectively. Then \(\mathrm{ABCD}\) is a (a) parallelogram but not a rhombus (b) square (c) rhombus (d) rectangle.

If \(\left|\begin{array}{lll}a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0 \quad\) 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 abc equals (a) 0 (b) 2 (c) \(-1\) (d) 1

The vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) \& \(\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\). The length of the median through A is (a) \(\sqrt{288}\) (b) \(\sqrt{18}\) (c) \(\sqrt{72}\) (d) \(\sqrt{33}\)

If \(\vec{a}\) and \(\vec{b}\) are unit vectors, then the greatest value of \(\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|\) is

If \(\underset{\rightarrow}{x}\) and \(y\) be two \(\underset{\rightarrow}{\rightarrow}\) non-zero vectors such that \(|x+y|=|x|\) and \(2 x+\lambda y\) is perpendicular to \(y\), then the value of \(\lambda\) is \(\quad\) [NA Sep. 06, 2020 (II)]

Let the vectors \(\vec{a}, \vec{b}, \vec{c}\) be such that \(|\vec{a}|=2,|\vec{b}|=4\) and \(|\vec{c}|=4\). If the projection of \(\vec{b}\) on \(\vec{a}\) is equal to the projection of \(\vec{c}\) on \(\vec{a}\) and \(\vec{b}\) is perpendicular to \(\vec{c}\), then the value of \(|\vec{a}+\vec{b}-\vec{c}|\) is

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8\). Then \(|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}\) is equal to \(\quad\) [NA Sep. 02, 2020 (I)]

The projection of the line segment joining the points \((1,-1,3)\) and \((2,-4,11)\) on the line joining the points \((-1,2,3)\) and \((3,-2,10)\) is \(\quad\). [NA Jan. 9, \(\mathbf{2 0 2 0}\) (I)]

Let the volume of a parallelopiped whose coterminous edges are given by \(\vec{u}=\hat{i}+\hat{j}+\lambda \hat{k}, \vec{v}=\hat{i}+\hat{j}+3 \hat{k}\) and \(\vec{w}=2 \hat{i}+\hat{j}+\hat{k}\) be \(1 \mathrm{cu}\). unit. If \(\theta\) be the angle between the edges \(\vec{u}\) and \(\vec{w}\), then \(\cos \theta\) can be: \(\quad\) [Jan. 8,2020 (I)] (a) \(\frac{7}{6 \sqrt{6}}\) (b) \(\frac{7}{6 \sqrt{3}}\) (c) \(\frac{5}{7}\) (d) \(\frac{5}{3 \sqrt{3}}\)

A vector \(\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in \boldsymbol{R})\) lies in the plane of the vectors, \(\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\hat{i}-\hat{j}+4 \hat{k}\). If \(\vec{a}\) bisects the angle between \(\vec{b}\) and \(\vec{c}\), then: [Jan. 7, 2020 (I)] (a) \(\vec{a} \cdot \hat{i}+3=0\) (b) \(\vec{a} \cdot \hat{i}+1=0\) (c) \(\vec{a} \cdot \hat{k}+2=0\) (d) \(\vec{a} \cdot \hat{k}+4=0\)

Let \(\vec{a}=2 \hat{i}+\lambda_{1} \hat{j}+3 \hat{k}, \vec{b}=4 \hat{i}+\left(3-\lambda_{2}\right) \hat{j}+6 \hat{k}\) and \(\overrightarrow{\mathrm{c}}=3 \hat{i}+6 \hat{j}+\left(\lambda_{3}-1\right) \hat{k}\) be three vectors such that \(\overrightarrow{\mathrm{b}}=2 \overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{a}}\) is perpendicular to \(\overrightarrow{\mathrm{c}}\) Then a possible value of \(\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)\) is: [Jan. \(\mathbf{1 0}, \mathbf{2 0 1 9}\) (I)] (a) \((1,3,1)\) (b) \(\left(-\frac{1}{2}, 4,0\right)\) (c) \(\left(\frac{1}{2}, 4,-2\right)\) (d) \((1,5,1)\)

Let \(\vec{a}=\hat{i}+\hat{j}+\sqrt{2} \hat{k}, \vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+\sqrt{2} \hat{k}\) and \(\overrightarrow{\mathrm{c}}=5 \hat{i}+\hat{j}+\sqrt{2} \hat{k}\) be three vectors such that the projection vector of \(\overrightarrow{\mathrm{b}}\) on \(\vec{a}\) is \(\vec{a}\). If \(\vec{a}+\vec{b}\) is perpendicular to \(\overrightarrow{\mathrm{c}}\), then \(|\overrightarrow{\mathrm{b}}|\) is equal to: [Jan. 09, 2019 (II)] (a) \(\sqrt{32}\) (b) 6 (c) \(\sqrt{22}\) (d) 4

If \(\hat{\mathrm{x}}, \hat{\mathrm{y}}\) and \(\hat{\mathrm{z}}\) are three unit vectors in three-dimensional space, then the minimum value of \(|\hat{\mathrm{x}}+\hat{\mathrm{y}}|^{2}+|\hat{\mathrm{y}}+\hat{\mathrm{z}}|^{2}+|\hat{\mathrm{z}}+\hat{\mathrm{x}}|^{2} \quad[\) Online April \(\mathbf{1 2}, \mathbf{2 0 1 4}]\) (a) \(\frac{3}{2}\) (b) 3 (c) \(3 \sqrt{3}\) (d) 6

If \(|\vec{a}|=2,|\vec{b}|=3\) and \(|2 \vec{a}-\vec{b}|=5\), then \(|2 \vec{a}+\vec{b}|\) equals: [Online April 9, 2014] (a) 17 (b) 7 (c) 5 (d) 1

If \(\hat{a}, \hat{b}\) and \(\hat{c}\) are unit vectors satisfying \(\hat{a}-\sqrt{3} \hat{b}+\hat{c}=\overrightarrow{0}\), then the angle between the vectors \(\hat{a}\) and \(\hat{c}\) is: |Online April 22, 2013] (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{6}\) (d) \(\frac{\pi}{2}\)

Let \(\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}+\hat{j}-2 \hat{k}\) be three vectors. A vector of the type \(\vec{b}+\lambda \vec{c}\) for some scalar \(\lambda\), whose projection on \(\vec{a}\) is of magnitude \(\sqrt{\frac{2}{3}}\) is : |Online April 9, 2013] (a) \(2 \hat{i}+\hat{j}+5 \hat{k}\) (b) \(2 \hat{i}+3 \hat{j}-3 \hat{k}\) (c) \(2 \hat{i}-\hat{j}+5 \hat{k}\) (d) \(2 \hat{i}+3 \hat{j}+3 \hat{k}\)

Let \(A B C D\) be a parallelogram such that \(\overrightarrow{A B}=\vec{q}, \overline{A D}=\vec{p}\) and \(\angle B A D\) be an acute angle. If \(\vec{r}\) is the vector that coincide with the altitude directed from the vertex \(\mathrm{B}\) to the side \(A D\), then \(\vec{r}\) is given by : (a) \(\vec{r}=3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (b) \(\vec{r}=-\vec{q}+\frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (c) \(\vec{r}=\vec{q}-\frac{(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (d) \(\vec{r}=-3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\)

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors. If the vectors \(\vec{c}=\hat{a}+2 \hat{b}\) and \(\vec{d}=5 \hat{a}-4 \hat{b}\) 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) \(\frac{\pi}{4}\)

If \(a+b+c=0,|\vec{a}|=3,|\vec{b}|=5\) and \(|\vec{c}|=7\), then the angle between \(\vec{a}\) and \(\vec{b}\) is [Online May 19, 2012] (a) \(\frac{\pi}{3}\) (b) \(\frac{\pi}{4}\) (c) \(\frac{\pi}{6}\) (d) \(\frac{\pi}{2}\)

A unit vector which is perpendicular to the vector \(2 \hat{i}-\hat{j}+2 \hat{k}\) and is coplanar with the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(2 \hat{i}+2 \hat{j}-\hat{k}\) is \([\) Online May 12, 2012] (a) \(\frac{2 \hat{j}+\hat{k}}{\sqrt{5}}\) (b) \(\frac{3 \hat{i}+2 \hat{j}-2 \hat{k}}{\sqrt{17}}\) (c) \(\frac{3 \hat{i}+2 \hat{j}+2 \hat{k}}{\sqrt{17}}\) (d) \(\frac{2 \hat{i}+2 \hat{j}-2 \hat{k}}{3}\)

\(A B C D\) is parallelogram. The position vectors of \(A\) and \(C\) are respectively, \(3 \hat{i}+3 \hat{j}+5 \hat{k}\) and \(\hat{i}-5 \hat{j}-5 \hat{k}\). If \(M\) is the midpoint of the diagonal \(D B\), then the magnitude of the projection of \(\overrightarrow{O M}\) on \(\overrightarrow{O C}\), where \(O\) is the origin, is [Online May 7, 2012] (a) \(7 \sqrt{51}\) (b) \(\frac{7}{\sqrt{50}}\) (c) \(7 \sqrt{50}\) (d) \(\frac{7}{\sqrt{51}}\)

If the vectors \(\vec{a}=\hat{i}-\hat{j}+2 \hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}+\hat{\hat{k}}\) and \(\vec{c}=\lambda \hat{i}+\hat{j}+\mu \hat{k}\) are mutually orthogonal, then \((\lambda, \mu)=\) \(\begin{array}{ll}\text { (a) }(2,-3) & \text { (b) }(-2,3)\end{array}\) [2010] (c) \((3,-2)\) (d) \((-3,2)\)

The non-zero vectors are \(\vec{a}, \vec{b}\) 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 \(\\{2008]\) (a) 0 (b) \(\frac{\pi}{4}\) (c) \(\frac{\pi}{2}\) (d) \(\pi\)

The non-zero vectors are \(\vec{a}, \vec{b}\) and \(c\) are related by \(\vec{a}=8 \vec{b}\) and \(\vec{c}=-7 \bar{b}\). Then the angle between \(\vec{a}\) and \(\vec{c}\) is [2008] (a) 0 (b) \(\frac{\pi}{4}\) (c) \(\frac{\pi}{2}\) (d) \(\pi\)

The values of a, for which 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 \(a \hat{i}-3 \hat{\jmath}+\hat{k}\) respectively are the vertices of a right angled triangle with \(C=\frac{\pi}{2}\) are \([2006]\) (a) 2 and 1 (b) \(-2\) and \(-1\) (c) \(-2\) and 1 (d) 2 and \(-1\)

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 \(|\bar{u}-\bar{v}+\bar{w}|\) equals (a) 14 (b) \(\sqrt{7}\) (c) \(\sqrt{14}\) (d) 2

\(\vec{a}, \vec{b}, \vec{c}\) are 3 vectors, such that \(\vec{a}+\vec{b}+\vec{c}=0\), \(|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3\), then \(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) is equal to \([2003]\) (a) 1 (b) 0 (c) \(-7\) (d) 7