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 are asked to find the dot product of \( \vec{a} \) with the expression \( (\vec{c} - 2\vec{b}) \). Using the distributive property of the dot product, this is equal to \( \vec{a} \cdot \vec{c} - 2(\vec{a} \cdot \vec{b}) \).

Step 2: Key Formula or Approach:
1. Distributive property: \( \vec{a} \cdot (\vec{c} - 2\vec{b}) = \vec{a} \cdot \vec{c} - 2\vec{a} \cdot \vec{b} \). 2. Use the given values: \( \vec{a} \cdot \vec{c} = 3 \). 3. Calculate \( \vec{a} \cdot \vec{b} \).

Step 3: Detailed Explanation:
1. Calculate \( \vec{a} \cdot \vec{b} \): \[ \vec{a} \cdot \vec{b} = (1\hat{i} + 1\hat{j} + 1\hat{k}) \cdot (0\hat{i} + 1\hat{j} - 1\hat{k}) \] \[ \vec{a} \cdot \vec{b} = (1)(0) + (1)(1) + (1)(-1) = 0 + 1 - 1 = 0 \] 2. Since \( \vec{a} \cdot \vec{b} = 0 \), the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular. 3. Substitute the values into the required expression: \[ \vec{a} \cdot (\vec{c} - 2\vec{b}) = \vec{a} \cdot \vec{c} - 2(\vec{a} \cdot \vec{b}) \] \[ = 3 - 2(0) = 3 \]

Step 4: Final Answer:
The value of \( \vec{a} \cdot (\vec{c} - 2\vec{b}) \) is 3.