Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k} \quad a_i>0, i=1,2,3$ be a vector which makes equal angles with the coordinates axes OX, OY and OZ . Also,...

Let $\hat{a}, \hat{b}$ be unit vectors. If $\vec{c}$ be a vector such that the angle between $\hat{a}$ and $\hat{c}$ is $\frac{\pi}{12}$, and $\hat{b}=\vec{c}+2(\vec{c} \times \hat{a})$,...

Let $P_1$ be the plane $3 x-y-7 z=11$ and $P_2$ be the plane passing through the points $(2,-1,0),(2,0,-1)$, and $(5,1,1)$. If the foot of the...

Let the vectors $\overrightarrow{\mathrm{u}}_1=\hat{\mathrm{i}}+\hat{\mathrm{j}}+a \hat{\mathrm{k}}, \overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+b \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{u}}_3=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be coplanar. If the vectors $\vec{v}_1=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \quad \vec{v}_2=a \hat{i}+(b+c) \hat{j}+a \hat{k}$ and $...

Let the area of the triangle formed by the lines $x+2=y-1=z, \frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1}$ and $\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1}$ be $A$. Then $A^2$ is equal to $\_\_\_\_$ .

For $\mathrm{a}, \mathrm{b} \in \mathrm{Z}$ and $|\mathrm{a}-\mathrm{b}| \leq 10$, let the angle between the plane $\mathrm{P}: \mathrm{ax}+\mathrm{y}-\mathrm{z}=\mathrm{b}$ and the line $l: \mathrm{x}-1=\mathrm{a} -y=z+1$ be $...

Let the foot of perpendicular from the point $\mathrm{A}(4,3,1)$ on the plane $\mathrm{P}: \mathrm{x}-\mathrm{y}+2 \mathrm{z}+3=0$ be N . If $\mathrm{B}(5, \alpha$, $\beta$ ), $...

The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(–2, –3, 5) and D(1, –6, –7) is equal to

Let the line $=\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$ at the points $A$ and $B$ respectively. Then the distance of the mid-point of the...

If the lines and intersect, then the magnitude of the minimum value of 8is _________.