The question in the image asks: "Which of the following represents the angle between the vectors $\vec{i} + \vec{j} + \vec{k}$ and $\vec{i} + \vec{j} - \vec{k}$?"

The approach provided uses the formula for the angle between two vectors $\vec{a}$ and $\vec{b}$ in three dimensions:

$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$

Step-by-step process:

1. Dot product $\vec{a} \cdot \vec{b}$: $\vec{a} = \vec{i} + \vec{j} + \vec{k}, \quad \vec{b} = \vec{i} + \vec{j} - \vec{k}$ The dot product of these two vectors is: $\vec{a} \cdot \vec{b} = (1 \cdot 1) + (1 \cdot 1) + (1 \cdot (-1)) = 1 + 1 - 1 = 1$

2. Magnitude of the vectors $|\vec{a}|$ and $|\vec{b}|$: The magnitude of both vectors is the same: $|\vec{a}| = |\vec{b}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$

3. Cosine of the angle $\theta$: Using the dot product and the magnitudes: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \frac{1}{\sqrt{3} \cdot \sqrt{3}} = \frac{1}{3}$

4. Angle $\theta$: Therefore, the angle is: $\theta = \cos^{-1}\left(\frac{1}{3}\right)$

Correct Answer:

The answer is Option B: $\cos^{-1}\left( \frac{1}{3} \right)$.

Would you like further explanation or more details?

Follow-up Questions:

1. How do you compute the dot product of two vectors in 3D?
2. How is the magnitude of a vector determined?
3. What does the angle between two vectors represent geometrically?
4. Why is the cosine formula used to find the angle between vectors?
5. How can the result $\cos^{-1}(1/3)$ be converted into degrees?

Tip:

When computing the angle between vectors, always ensure that the magnitudes of the vectors are correctly calculated before using the cosine formula.