Let (α,β,γ) be the foot of perpendicular from the point (1,2,3) on the line (x+3)/5=(y-1)/2=(z+4)/3. then 19(α+β+γ) is equal to :

Let O be the origin and the position vector of A and B be $2 \hat{\imath}+2 \hat{\jmath}+\hat{k}$ and $2 \hat{\imath}+4 \hat{\jmath}+4 \hat{k}$ respectively. If the...

Let $\vec{a}=a_i \hat{\imath}+a_2 \hat{\jmath}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{\imath}+b_2 \hat{\jmath}+b_3 \hat{k}$ be two vectors such that $|\vec{a}|=1 ; \vec{a} \cdot \vec{b}=2$ and $|\vec{b}|=4$. If $...

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{\imath}-2 \hat{\jmath}+2 k$ and $\alpha \hat{\imath}+ 2 \alpha \hat{\jmath}-2 k$...

Let $\overrightarrow{\mathrm{a}}=\hat{\imath}+2 \hat{\jmath}+\mathrm{k}, \overrightarrow{\mathrm{b}}=3(\hat{\imath}-\hat{\jmath}+\mathrm{k})$. Let $\vec{c}$ be the vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$. Then $\vec{a} \cdot((\vec{c} \times \vec{b})-\vec{b}-\vec{c})$ is equal...

If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac{6}{\sqrt{5}}$, then the sum of all possible values of $\lambda$ is :

The distance, of the point (7,-2,11) from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :

Let $\hat{\mathrm{a}}$ be a unit vector perpendicular to the vectors $\overrightarrow{\mathrm{b}}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\overrightarrow{\mathrm{c}}=2 \hat{i}+3 \hat{j}-\hat{k}$, and makes an angle of $\cos ^{-1}\left(-\frac{1}{3}\right)$ with...

Let P be the foot of the perpendicular from the point $(1,2,2)$ on the line $\mathrm{L}: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$. Let the line $\overrightarrow{\mathrm{r}}=(-\hat{i}+\hat{j}-2 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}), \lambda \in \mathbf{R}$,...