https://file.aiquickdraw.com/mathbot/file/1726351988478v08tzmty.jpg

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}.


Which of the following represents the angle between the vectors i + j + k and i + j - k?

Learn how to find the angle between the vectors i + j + k and i + j - k using the dot product formula. This problem explores vector mathematics, including the use of cosine to find the angle between two vectors in 3D space. Suitable for students in advanced high school or college-level math.

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