Let's analyze the problem presented in the image:

You are given two vectors: $\mathbf{a} = \hat{i} + \hat{j} + \hat{k}$ and $\mathbf{b} = \hat{i} + \hat{j} - \hat{k}.$ The question asks for the angle between these two vectors. To find the angle $\theta$, we use the formula for the dot product of two vectors: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta.$

Step 1: Calculate the dot product $\mathbf{a} \cdot \mathbf{b}$

The dot product is calculated as: $\mathbf{a} \cdot \mathbf{b} = (1)(1) + (1)(1) + (1)(-1) = 1 + 1 - 1 = 1.$

Step 2: Calculate the magnitudes of $\mathbf{a}$ and $\mathbf{b}$

The magnitude of a vector $\mathbf{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}$ is given by: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}.$ For $\mathbf{a}$, we have: $|\mathbf{a}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}.$ For $\mathbf{b}$, we have: [ |\mathbf{b}| = \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{3}.