Question

If the coordinates of the point where the line $x-2y+z-1=0=x+2y-2z-5$ intersects the plane $x+y-2z=7$ is $\left(\right.\text{α},\text{β},\text{γ}\left.\right),$ then find the value of $\left(\right.\left|\right.\text{α}\left|\right.+\left|\right.\text{β}\left|\right.+\left|\right.\text{γ}\left|\right.\left.\right)$

Answer 1

The required point is the point of intersection of the three planes.
$x+y-2z=7.......\left(\right.1\left.\right)$
$x-2y+z=1.......\left(\right.2\left.\right)$
$x+2y-2z=5........\left(\right.3\left.\right)$
$\therefore $ From $\left(\right.3\left.\right)-\left(\right.1\left.\right)\Rightarrow y=-2$
From $2\times \left(\right.2\left.\right)+\left(\right.1\left.\right)\Rightarrow 3x-3y=9\Rightarrow x=1$
So, from $\left(\right.1\left.\right),z=-4$ Hence the point is $\left(\right.1,-2,-4\left.\right)=\left(\right.\text{α},\text{β},\text{γ}\left.\right)$ Hence $\left|\right.\text{α}\left|\right.+\left|\right.\text{β}\left|\right.+\left|\right.\text{γ}\left|\right.$
$=\left|\right.1\left|\right.+\left|\right.-2\left|\right.+\left|\right.-4\left|\right.=7$