Question

If P = (0, 1, 0), Q =(0, 0, 1), then projection of PQ on the plane $x + y + z = 3$ is

• A. $\sqrt { 3 }$
• B. 3
• C. $\sqrt { 2 }$
• D. 2

Answer 1

Answer: (C)

$\sqrt { 2 }$

Answer 2

Given plane is $x + y + z - 3 = 0$ . From point P and Q draw PM and QN perpendicular on the given plane and QR ⊥ MP.

$| M P | = \left| \frac { 0 + 1 + 0 - 3 } { \sqrt { 1 ^ { 2 } + 1 ^ { 2 } + 1 ^ { 2 } } } \right| = \frac { 2 } { \sqrt { 3 } }$

$| N Q | = \frac { 2 } { \sqrt { 3 } }$ $| P Q | = \sqrt { ( 0 - 0 ) ^ { 2 } + ( 0 - 1 ) ^ { 2 } + ( 1 - 0 ) ^ { 2 } } = \sqrt { 2 }$

$| R P | = | M P | - | M R | = | M P | - | N Q | = 0$

(i.e. R and P are the same point)

∴ $| N M | = | Q R | = \sqrt { P Q ^ { 2 } - R P ^ { 2 } } = \sqrt { ( \sqrt { 2 } ) ^ { 2 } - 0 } = \sqrt { 2 }$