To find the value of \( \alpha \) such that the projection of vector \( \mathbf{a} = 2\hat{i} + 4\hat{j} - 2\hat{k} \) on vector \( \mathbf{b} = \hat{i} + 2\hat{j} - \alpha\hat{k} \) is zero, we start by recalling the formula for the projection of \( \mathbf{a} \) onto \( \mathbf{b} \):
$$ \text{Proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}. $$
For this projection to be zero, the numerator must be zero:
$$ \mathbf{a} \cdot \mathbf{b} = 0. $$
Calculate the dot product:
$$ \mathbf{a} \cdot \mathbf{b} = (2)(1) + (4)(2) + (-2)(-\alpha) = 2 + 8 + 2\alpha = 10 + 2\alpha. $$
Set the dot product equal to zero:
$$ 10 + 2\alpha = 0. $$
Solving for \( \alpha \):
$$ 2\alpha = -10 $$
$$ \alpha = -5. $$
Verify the range:
The given range is 5, 5, but the computed solution \( \alpha = -5 \) does not fall within this range. It appears there might be a misinterpretation or typo in the range, as the mathematical solution holds true. Therefore, the correct value of \( \alpha \) that results in a zero projection is clearly \( -5 \).
