Question

Let \( \vec{a} = \hat{i} - \hat{j} \), \( \vec{b} = \hat{j} - \hat{k} \), and \( \vec{c} = \hat{k} - \hat{i} \), then the value of \( \vec{b} \cdot (\vec{a} + \vec{c}) \) is:

• A. 1
• B. 0
• C. -1
• D. 2
• E. -2

Hint:

Use the distributive property of dot products and substitute vector components to solve.

Answer 1

Answer: (E)

We are asked to find \( \vec{b} \cdot (\vec{a} + \vec{c}) \). Using the distributive property of the dot product: \[ \vec{b} \cdot (\vec{a} + \vec{c}) = \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} \] Now substitute the components of \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) into the dot product and compute: \[ \vec{b} \cdot \vec{a} = 1 \times (-1) + (-1) \times 1 = -2 \] Thus, the correct answer is \( -2 \).