A unit vector a makes an angle `pi/ 4` with z-axis. If `a + i + j` is a unit vector, then a can be equal to

Explanation

To solve the problem step by step, we need to find the vector \( \mathbf{a} \) given that it makes an angle of \( \frac{\pi}{4} \) with the z-axis and that \( \mathbf{a} + \mathbf{i} + \mathbf{j} \) is a unit vector.

### Step 1: Define the vector \( \mathbf{a} \)
Let \( \mathbf{a} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \).

### Step 2: Use the condition of being a unit vector
Since \( \mathbf{a} \) is a unit vector, we have:
\[
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