Amit M Agarwal Solutions for Chapter: Three Dimensional Coordinate System, Exercise 5: Three Dimensional Coordinate System Exercise 2

Consider the planes $P_1: 2x+y+z+4=0, P_2: y-z+4=0$ and $P_3: 3x+2y+z+8=0.$ Let $L_1,L_2,L_3$ be the lines of intersection of the planes $P_2 \& P_3, P_{3 }\& P_1$, and $P_1 \& P_2$, respectively. Then,

The volume of a right triangular prism $ABC{A}_1{B}_1{C}_1$ is equal to $3$cubic unit. Find the coordinates of the vertex $A_1$, if the coordinates of the base vertices of the prism are $A(1,0,1),B(2,0,0)$ and $C(0,1,0)$.

Let a plane pass through origin and is parallel to the line $\frac{x–1}{2}=\frac{y+3}{–1}=\frac{z+1}{–2}$ such that distance between plane and the line is $\frac{5}{3}$ Then, equation of the plane is

$OABC$ is a regular tetrahedron of side unity, then

If $OABC$ is a tetrahedron such that ${\left|\overrightarrow{OA}\right|}^2+{\left|\overrightarrow{BC}\right|}^2={\left|\overrightarrow{OB}\right|}^2+{\left|\overrightarrow{CA}\right|}^2={\left|\overrightarrow{OC}\right|}^2+{\left|\overrightarrow{AB}\right|}^2$, then

If the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ intersects the line $3{\beta }^{2}x+3\left(1-2\alpha \right)y+z=3=-\frac{1}{2}\left\{6{\alpha }^{2}x+3\left(1-2\beta \right)y+2z\right\}$ then point $\left(\alpha ,\beta ,1\right)$ lie on the plane

Let $PM$ be the perpendicular from the point $P(1,2,3)$ to $XY$ plane. If $\overrightarrow{OP}$ makes an angle $\theta$ with the positive direction of $Z$-axis and $\overrightarrow{OM}$ makes an angle $\varphi$ with the positive direction of $X$-axis, where $O$ is the origin and $\theta$ and $\varphi$ are acute angles, then

A variable plane which remains at a constant distance $p$ from the origin $\left(0\right)$ cuts the coordinate axes in $A,B,C$