The set of all $\alpha$, for which the vector $\vec{a}=\alpha t \hat{\imath}+6 \hat{\jmath}-3 \hat{k}$ and $\vec{b}=t \hat{\imath}-2 \hat{\jmath}-2 \alpha t \hat{k}$ are inclined at an...

If A(1,-1,2),B(5,7,-6),C(3,4,-10) and D(-1,-4,-2) are the vertices of a quadrilateral ABCD, then its area is

Let the point, on the line passing through the points $\mathrm{P}(1,-2,3)$ and $\mathrm{Q}(5,-4,7)$, farther from the origin and at a distance of 9 units from...

Let the line L intersect the lines x-2=-y=z-1,2(x+1)=2(y-1)=z+1 and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following points lies on L ?

The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is $\_\_\_\_$

Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{\imath}+2 \hat{\jmath}-\hat{k}$ and an angle of $45^{\circ}$ with $\hat{\imath}-\hat{k}$ be $\vec{C}$. Then $...

If the shortest distance between the lines L_1:r ⃗=(2+λ)i ˆ+(1-3λ)j ˆ+(3+4λ)k ˆ,λ∈R L_2:r ⃗=2(1+μ)i ˆ+3(1+μ)j ˆ+(5+μ)k ˆ,μ∈R is m/√n, where gcd⁡(m,n)=1, then the value of...

Let $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ be a point in the first octant, whose projection in the xy -plane is the point Q . Let $\mathrm{OP}=\gamma$; the...

If $\mathrm{A}(3,1,-1), \mathrm{B}\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), \mathrm{C}(2,2,1)$ and $\mathrm{D}\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral ABCD , then its area is $\_\_\_\_$ .

Let $\vec{a}, \vec{b}$ and $\overrightarrow{\mathrm{c}}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear if $\vec{a}+5 \vec{b}$is collinear with$\vec{c}, \vec{b}+6 \vec{c}$ is collinear...