Question

Given vectors \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{b} = \hat{j} - \hat{k} \). Another vector \( \vec{c} \) satisfy equations \( \vec{a} \cdot \vec{c} = 3 \) and \( \vec{a} \times \vec{c} = \vec{b} \), then find \( \vec{a} \cdot (\vec{c} - 2\vec{b}) \):

Answer 1

Correct Answer: 3

Step 1: Understanding the Concept:
We use the linearity of the dot product: \( \vec{a} \cdot (\vec{c} - 2\vec{b}) = \vec{a} \cdot \vec{c} - 2(\vec{a} \cdot \vec{b}) \). The value of \( \vec{a} \cdot \vec{c} \) is given, and \( \vec{a} \cdot \vec{b} \) can be calculated from the given vectors.
Step 2: Key Formula or Approach:
1. Dot product calculation: \( \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z \). 2. Distributive law: \( \vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} \).
Step 3: Detailed Explanation:
1. Calculate \( \vec{a} \cdot \vec{b} \): \( \vec{a} = (1, 1, 1) \), \( \vec{b} = (0, 1, -1) \). \( \vec{a} \cdot \vec{b} = (1)(0) + (1)(1) + (1)(-1) = 0 + 1 - 1 = 0 \). 2. Substitute into the expression: \( \vec{a} \cdot (\vec{c} - 2\vec{b}) = \vec{a} \cdot \vec{c} - 2(\vec{a} \cdot \vec{b}) \). Since \( \vec{a} \cdot \vec{c} = 3 \) and \( \vec{a} \cdot \vec{b} = 0 \): \( 3 - 2(0) = 3 \).
Step 4: Final Answer:
The result is 3.