Dot Product of Two Vectors

Two projectiles are fired from the same point with the same speed at angles of projection $60°\mathrm{and}30°$ respectively. The correct statement is

If $\overrightarrow{a}\perp \overrightarrow{b}$ and $(\overrightarrow{a}+\overrightarrow{b})\perp (\overrightarrow{a}+m\overrightarrow{b})$, then $m$ is

Force $\overrightarrow{F}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ is acting on a particle. If the particle is displaced from $A\left(3,3,3\right)$ to the point $B\left(4,4,4\right)$, then work done is

If $\overrightarrow{e_1} \& \overrightarrow{e_2}$ are two unit vectors and $\theta$ is the angle between them, then $\sin \left(\frac{\theta }{2}\right)$ is -

Let $\hat{i}, \hat{j}$ and $\hat{k}$ denote the standard unit vectors in $R^3$ along the $x$-axis, $y$-axis and $z$-axis, respectively. Consider the sets

$X=\left\{a\hat{i}+b\hat{j}+c\hat{k} : a,b,c\in \left\{-1,0,1\right\}\right\}$ and

$Y=${$\left(\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\right) :\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\in X$ and $\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}$ are mutually perpendicular unit vectors}

Then, the number of elements in $Y$ is

Let $\overrightarrow{A}=\hat{i}+2\hat{j}.$ If $\overrightarrow{B}$ is a vector in $XY$ plane such that $(\overrightarrow{A}+\overrightarrow{B})·\overrightarrow{B}=15$ and $\overrightarrow{A}·\overrightarrow{B}=6,$ then $\left|\overrightarrow{B}\right|$ is

If $\overrightarrow{a}=x^2\hat{i}+x\hat{j}+3\hat{k}$ and $\overrightarrow{b}=x\hat{i}-4\hat{j}+2\hat{k}$ and $\overrightarrow{a}·\overrightarrow{b}>6$, then

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three vectors such that $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4, \left|\overrightarrow{c}\right|=5$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0,$ then $\overrightarrow{a}·\overrightarrow{b}$ is equal to

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}, \overrightarrow{d}$ are vectors in which $\left|\overrightarrow{d}\right|=1$ and given $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=s\overrightarrow{d}, \overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}=\overrightarrow{a},$ $\overrightarrow{a}·\overrightarrow{d}=4,$ then ${}^{\text{'}}s^{\text{'}}$ is equal to

The values of $x$, for which the vectors $\overrightarrow{a}=\left(x^2-16\right)\hat{i}+4\hat{j}-\left(x^2-25\right)\hat{k}$ make acute angles with the (positive directions of) coordinate axes lie in

If $\left|\overrightarrow{x}\right|=\left|\overrightarrow{y}\right|=\left|\overrightarrow{x}+\overrightarrow{y}\right|=1,$ then $\left|\overrightarrow{x}-\overrightarrow{y}\right|=\_\_\_\_\_\_\_$

The perpendicular distance of point $\left(\mathrm{3,4},5\right)$ on the line $\overrightarrow{r}=\left(2\widehat{i}+3\widehat{j}+4\widehat{k}\right)+\lambda \left(2\widehat{i}+3\widehat{j}+4\widehat{k}\right)$ is