If the projection of 2 ^ i + 4 ^ j - 2 ^ k on ^ i + 2 ^ j + α ^ k...

Given here:

\overset{\to}{a} = 2\overset{^}{i} + 4\overset{^}{j} - 2\overset{^}{k} and \overset{\to}{b} = \overset{^}{i} + 2\overset{^}{j} + \alpha\overset{^}{k}

Now, $\overset{\to}{a} \cdot \overset{\to}{b} = 0$, as $\overset{\to}{a}$ should be perpendicular to $\overset{\to}{b}$.

$∴ 2 \times 1 + 4 \times 2 - 2 \times \alpha = 0$

$∴ \alpha = 5$