Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$ and the angle between $\vec{b}$...

Let the line of the shortest distance between the lines $\mathrm{L}_1: \vec{r}=(\hat{\imath}+2 \hat{\jmath}+3 \hat{k})+\lambda(\hat{\imath}-\hat{\jmath}+\hat{k})$ and $\mathrm{L}_2: \overrightarrow{\mathrm{r}}= (4 \hat{\imath}+5 \hat{\jmath}+6 \hat{k})+\mu(\hat{\imath}+\hat{\jmath}-\hat{k})$ intersect $L_1$ and $L_2$...

Let $Q$ and $R$ be the feet of perpendiculars from the point $P(a, a, a)$ on the lines $x=y, z=1$ and $x=-y, z=$ -1 respectively....

If d_1 is the shortest distance between the lines x+1=2y=-12z,x=y+2=6z-6 and d_2 is the shortest distance between the lines (x-1)/2=(y+8)/(-7)=(z-4)/5,(x-1)/2=(y-2)/1=(z-6)/(-3), then the value of (32√3...

A line with direction ratios 2,1,2 meets the lines x=y+2=z and x+2=2y=2z respectively at the point P and Q. if the length of the perpendicular...

If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ is 1 , then the sum of all possible values of $\lambda$ is :

The distance of the point $Q(0,2,-2)$ form the line passing through the point $P(5,-4,3)$ and perpendicular to the lines $...

Let $\vec{a}=3 \hat{\imath}+\hat{\jmath}-2 \hat{k}, \vec{b}=4 \hat{\imath}+\hat{\jmath}+7 \hat{k}$ and $\overrightarrow{\mathrm{c}}=\hat{\imath}-3 \hat{\jmath}+4 \hat{k}$ be three vectors. If a vectors $\overrightarrow{\mathrm{p}}$ satisfies $\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{b}}= \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}$...

Let PQR be a triangle with R(-1,4,2). Suppose M(2,1,2) is the mid point of PQ. The distance of the centroid of △PQR from the point...