Vector Algebra

The work done by the force $4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ in moving a particle along a straight line from the point $(3,2,-1)$ to $(2,-1,4)$ is

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are three non-coplanar vectors, then $[\mathbf{a} \times \mathbf{b} \mathbf{b} \times \mathbf{c} \mathbf{c} \times \mathbf{a}]$ is equal to

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are any three mutually perpendicular vectors of equal magnitude $a$, then $|\mathbf{a}+\mathbf{b}+\mathbf{c}|$

If $|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|$, then $\mathbf{a}$ and $\mathbf{b}$ are

If $\mathbf{a}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{c}=-4 \hat{\mathbf{i}}-\hat{\mathbf{k}}$, then $\mathbf{a}…

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are non-collinear vectors such that for some scalars, $x, y, z$, $x \mathbf{a}+x \mathbf{b}+x \mathbf{c}=0$, then

Three concurrent edges of a parallelopiped are given by $\begin{aligned} \mathbf{a} & =2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \\ \mathbf{b} & =\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}} \\…

If $\mathbf{a}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $c=-4 \hat{i}-\hat{k}$, then $\mathbf{a} \times(b \times c)+(a \cdot b) c$

The value of $|\vec{a} \times \vec{b}+\vec{b} \times \vec{a}|$ is

A parallelogram is constructed on the vectors $\mathbf{a}=3 \alpha-\beta, \mathbf{b}=\alpha+3 \beta$. If $|\alpha|=|\beta|=2$ and the angle between $\alpha$ and $\beta$ is $\frac{\pi}{3}$, then length of a diagonal of…

If $\left|\begin{array}{lll}a & a^2 & 1+a^3 \\ b & b^2 & 1+b^3 \\ c & c^2 & 1+c^3\end{array}\right|=0$ 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 number of vectors of unit length perpendicular to vectors $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\mathbf{b}=\hat{\mathbf{j}}+\hat{\mathbf{k}}$, is