Let $P(3,2,3)$, $Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\triangle PQR$. Then, the angle $\angle QPR$ is
Let a line passing through the point $(-1,2,3)$ intersect the lines $L_1:\dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z+1}{-2}$ at $M(\alpha,\beta,\gamma)$ and $L_2:\dfrac{x+2}{-3}=\dfrac{y-2}{-2}=\dfrac{z-1}{4}$ at $N(a,b,c)$. Then the value of $\dfrac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals
Let a line pass through two distinct points P (-2, -1, 3) and Q, and be parallel to the vector 3^i + 2^j + 2k ^ . If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △P QR is equal to :
A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a,b,c)$ where $a,b,c$ are non-negative integers, on the line $AB$ lies at a distance of 21 units, from the point $A$. The distance between the points $P(a,b,c)$ and $Q(4,-12,3)$ is equal to
Let $L_1:\dfrac{x-1}{3}=\dfrac{y-1}{-1}=\dfrac{z+1}{0}$ and $L_2:\dfrac{x-2}{2}=\dfrac{y}{0}=\dfrac{z+4}{\alpha}$, $\alpha\in\mathbf{R}$, be two lines which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1,1,-1)$ on $L_2$, then the value of $26\alpha(PB)^2$ is ________.
Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3\hat{i}+2\hat{j}+2\hat{k}$. If the distance of the point $Q$ from the point $R(1,3,3)$ is $5$, then the square of the area of $\triangle PQR$ is equal to:
Consider a line $L$ passing through the points $P(1,2,1)$ and $Q(2,1,-1)$. If the mirror image of the point $A(2,2,2)$ in the line $L$ is $(\alpha,\beta,\gamma)$, then $\alpha+\beta+6\gamma$ is equal to _____
The given line is \(x = 4y+5,\; z = 3y-6\). A point on the line is \((4\lambda+5,\;\lambda,\;3\lambda-6)\). The distance between the point \((4\lambda+5,\;\lambda,\;3\lambda-6)\) and \((5,3,-6)\) is 3 units. Find the point on the line closest to \((5,3,-6)\).