If two vectors → P = ^ i + 2m ^ j + m ^ k and → Q = 4 ^ i - 2 ^ j +...

For two vectors to be perpendicular, their dot product is equal to zero.

So, $\overset{\to}{P} \cdot \overset{\to}{Q} = 0$

⇒ \left(\overset{^}{i} + 2m\overset{^}{j} + m\overset{^}{k}\right) \cdot \left(4\overset{^}{i} - 2\overset{^}{j} + m\overset{^}{k}\right) = 0 ⇒ 4 - 4m + {m}^{2} = 0 ⇒ {\left(m - 2\right)}^{2} = 0 ⇒ m = 2