First slide

Planes in 3D

Question

The three planes $x + y = 0, y + z = 0 a n d x + z = 0$

Difficult

A

Meet in a unique point
B

Meet in a line
C

Meet taken two at a time in parallel lines
D

None of these

Solution

The planes $x + y = 0$ i.e. $x = - y$ and $y + z = 0$ i.e. $z = - y$ meet in the line $\frac{x}{1} = \frac{y}{- 1} = \frac{z}{1}$ .any point on this line is $(t, - t, t)$ .
This point lies in the plane $x + z = 0$ if $t + t = 0$ $\Rightarrow t = 0$ .

So the three planes meet in a unique point (0,0,0)

ALTERNATE:

The equation $x + y = 0, y + z = 0$ and $z + x = 0$ has no non trivial solution as $| \begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix} | \neq 0$ . So only $x = y = z = 0$ satisfies the given equations.

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