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: