Vector Algebra

A rigid body is rotating at the rate of $3$ radians per second about an axis $AB$, where $A(1,-2,1)$ and $B(3,-4,2)$. The velocity of the point $P(5,-1,-1)$ of the body is

Let $\vec A=2\vec i+\vec j-2\vec k$ and $\vec B=\vec i+\vec j$. If $\vec C$ is a vector such that $\vec A\cdot\vec C=|\vec C|$, $|\vec C-\vec A|=2\sqrt2$ and the angle between $\vec A\times\vec B$ and $\vec C$ is $30^\circ$, then $|(\vec A\times\vec B)\times\vec C|$ is equal to

The value of $\lambda$ such that the four points whose position vectors are $3\vec i-2\vec j+\lambda\vec k,\ 6\vec i+3\vec j+\vec k,\ 5\vec i+7\vec j+3\vec k$ and $2\vec i+2\vec j+6\vec k$ are coplanar is

The value of $\lambda$ for which the volume of the parallelepiped formed by the vectors $\vec i+\lambda\vec j+\vec k,\ \vec j+\lambda\vec k,\ \lambda\vec i+\vec k$ is minimum is

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar unit vectors and $\vec{a} \times (\vec{b} \times \vec{c}) = \dfrac{\vec{b} + \vec{c}}{\sqrt{2}}$, then the angle between $\vec{a}$ and $\vec{b}$ is:

The vector $\vec{B} = 3\vec{i} + 4\vec{k}$ is to be written as the sum of a vector $\vec{B_1}$ parallel to $\vec{A} = \vec{i} + \vec{j}$ and a vector $\vec{B_2}$ perpendicular to $\vec{A}$. Then $\vec{B_1}$ is:

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors, then $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2$ does not exceed:

$ABCD$ is a parallelogram with diagonals $AC$ and $BD$. Compute $ \overrightarrow{AC} - \overrightarrow{BD} $.

If $ |\vec{a}\times \vec{b}| = |\vec{a}\cdot \vec{b}| $, then angle $\theta$ between $\vec{a},\vec{b}$ is:

$ \vec{v} = 2\hat{i} + \hat{j} - \hat{k},\quad \vec{w} = \hat{i} + 3\hat{k} $ If $ \vec{u} $ is a unit vector, maximum value of $ [\vec{u}\ \vec{v}\ \vec{w}] $ is:

$ \vec{a} = x\hat{i} - 3\hat{j} - \hat{k},\quad \vec{b} = 2x\hat{i} + x\hat{j} - \hat{k} $ Angle between $ \vec{a} $ and $ \vec{b} $ is acute and angle between $ \vec{b} $ and $ +y $ axis lies in $ \left(\dfrac{\pi}{2}, \pi\right) $ Find $x$.

If $ a,b,c $ are coplanar, evaluate $ [,2a - b, 2b - c,2c - a,] $

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of three vertices A, B, C of a triangle respectively then the area of this triangle is given by

If there vectors $2\hat{i}-\hat{j}+\hat{k}$ , $\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda \hat{j}+5\hat{k}$ are coplanar, then $\lambda$ is

The area of the parallelogram whose diagonals are $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}-3\hat{j}+4\hat{k}$ is

Force $3\hat{i}+2\hat{j}+5\hat{k}$ and $2\hat{i}+\hat{j}-3\hat{k}$ are acting on a particle and displace it from the point $2\hat{i}-\hat{j}-3\hat{k}$ to $4\hat{i}-3\hat{j}+7\hat{k}$ the point then the work done by the force is

If the vector $2\hat{i}-3\hat{j}$ , $\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}-\hat{k}$ form three conterminous edges of a parallelepiped, then thevolume of parallelepiped is

Let $\vec{a}=\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ . Then the vector $\vec{b}$ satisfying $(\vec{a} \times \vec{b})+ \vec{c} =0$ and $\vec{a} . \vec{b}=3$ is

The sum of two vectors $\vec{a}$ and $\vec{b}$ is a vector $\vec{c}$ such that $|\vec{a}|=|\vec{b}|=|\vec{c}|=2$. Then, the magnitude of $\vec{a}-\vec{b}$ is equal to:

If $\vec{A}=4\hat{i}+3\hat{j}+\hat{k}$ and $\vec{B}=2\hat{i}-\hat{j}+2\hat{k}$ , then the unit vector $\hat{N}$ perpendicular to the vectors $\vec{A}$ and $\vec{B}$ ,such that $\vec{A}, \vec{B}$ , and $\hat{N}$ form a right handed system, is:

For the vectors $\vec{a}=-4\hat{i}+2\hat{j}, \vec{b}=2\hat{i}+\hat{j}$ and $\vec{c}=2\hat{i}+3\hat{j}$, if $\vec{c}=m\vec{a}+n\vec{b}$ then the value of m + n is

Constant forces $\vec{P}= 2\hat{i} - 5\hat{j} + 6\hat{k} $ and $\vec{Q}= -\hat{i} + 2\hat{j}- \hat{k}$  act on a particle. The work done when the particle is displaced from A whose position vector is $4\hat{i} - 3\hat{j} - 2\hat{k} $, to B whose position vector is $6\hat{i} + \hat{j} - 3k\hat{k}$ , is:

If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}|=13$, $|\vec{b}|=5$ and $\vec{a} . \vec{b} =60$then the value of $|\vec{a} \times \vec{b}|$ is

If $\vec{AC}=2\hat{i}+\hat{j}+\hat{k}$ and $\vec{BD}=-\hat{i}+3\hat{j}+2\hat{k}$ then the area of the quadrilateral ABCD is

If $\vec{a}, \vec{b}$ and $\vec{c}$ are the position vectors of the vertices A, B, C of a triangle ABC, then the area of the triangle ABC is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\frac{\vec{c}}{|\vec{c}|}$ is $\frac{1}{\sqrt{3}}$, is

If $\vec{a}=\hat{i}-\hat{k},\, \vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$ , then $[\vec{a} , \vec{b}, \vec{c}]$ depends on

Let $\vec{a}$ and $\vec{b}$ be two vectors, which of the following vectors are not perpendicular to each other?

If $\vec{a}$ and $\vec{b}$ in space, given by $\vec{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\vec{b}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}$ , then the value of $(2\vec{a}+\vec{b}).[(\vec{a} \times \vec{b}) \times (\vec{a}-2\vec{b})]$ is

If = (i + 2j - 3k) and =(3i -j + 2k), then the angle between ( + ) and ( - )

Let ,  and  be three vector such that || = 2, || = 3, || = 5 and ++ = 0. The value of .+.+. is

The value of non-zero scalars α and  β such that for all vectors $\vec{a}$  and $\vec{b}$ such that $\alpha (2\vec{a}-\vec{b})+\beta (\vec{a}+2\vec{b})=8\vec{b}-\vec{a}$ is

If the volume of a parallelepiped whose adjacent edges are 

Vertices of the vectors i - 2j + 2k , 2i + j - k and 3i - j + 2k form a triangle. This triangle is

If a, b, c are three non-zero vectors with no two of which are collinear, a + 2b  is collinear with c and b + 3c is collinear with a , then | a + 2b + 6c | will be equal to

The position vectors of points A and B are  and  . Then the position vector of point p dividing AB in the ratio m : n is

Forces of magnitude 5, 3, 1 units act in the directions 6i + 2j + 3k, 3i - 2j + 6k, 2i - 3j - 6k respectively on a particle which is displaced from the point (2, −1, −3) to (5, −1, 1). The total work done by the force is

If  are four vectors such that is collinear with  and is collinear with  then  =

Two forces F₁ and F₂ are used to pull a car, which met an accident. The angle between the two forces is θ . Find the values of θ for which the resultant force is equal to

If  are three non-coplanar vectors, then 

If $\vec{e_1}=(1,1,1)$ and $\vec{e_2}=(1,1,-1)$ and $\vec{a}$ and $\vec{b}$  and two vectors such that $\vec{e_2}=\vec{a}+2\vec{b}$ , then angle between $\vec{a}$ and $\vec{b}$

Angle between $\vec{a}$ and  $\vec{b}$ is $120{^{\circ}}$. If $|\vec{b}|=2|\vec{a}|$ and the vectors , $\vec{a}+x\vec{b}$ ,   $\vec{a}-\vec{b}$ are at right angle, then $x=$

Let $\vec{a}=\hat{i}+\hat{j}$ and  $\vec{b}=2\hat{i}-\hat{k}$, the point of intersection of the lines $\vec{r}\times\vec{a}=\vec{b}\times\vec{a}$  and  $\vec{r}\times\vec{b}=\vec{a}\times\vec{b}$  is

If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}$ , $(a,b,c\ne1)$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

The area of the triangle formed by the vertices whose position vectors are $3\widehat{i}+\widehat{j}$ , $5\widehat{i}+2\widehat{j}+\widehat{k}$ , $\widehat{i}-2\widehat{j}+3\widehat{k}$ is

If the position vector of A and B relative to O be $\widehat{i}\, -4\widehat{j}+3\widehat{k}$ and $-\widehat{i}\, +2\widehat{j}-\widehat{k}$ respectively, then the median through O of ΔABC is:

Let $\vec{a}=2\widehat{i}\, +\widehat{j}\, +2\widehat{k}$ , $\vec{b}=\widehat{i}-\widehat{j}+2\widehat{k}$ and $\vec{c}=\widehat{i}+\widehat{j}-2\widehat{k}$ are are three vectors. Then, a vector in the plane of $\vec{a}$ and $\vec{c}$ whose projection on $\vec{b}$ is of magnitude $\frac{1}{\sqrt{6}}$ is

If $\overrightarrow{{a}}$ and $\overrightarrow{{b}}$ are vectors in space, given by $\overrightarrow{{a}}=\frac{\hat{i}-2\hat{j}}{\sqrt[]{5}}$ and $\overrightarrow{{b}}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt[]{14}}$, then the value of$(2\vec{a} + \vec{b}).[(\vec{a} × \vec{b}) × (\vec{a} – 2\vec{b})]$ is