Question

Let \(\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}\), \(\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}\) and a vector \(\vec{c}\) be such that \(2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{c}) = 0\). If \(\vec{a} \cdot \vec{c} = 15\), then the value of \(\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})\) is

• A. 5
• B. -5
• C. 3
• D. -3

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We use vector algebra properties, specifically the cross product relation. The equation \(2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{c}) = 0\) can be rewritten to show a relationship between \(\vec{c}\) and \(\vec{a}\).

Step 2: Key Formula or Approach:
1. \(2(\vec{a} \times \vec{b}) - 3(\vec{c} \times \vec{b}) = 0 \implies (2\vec{a} - 3\vec{c}) \times \vec{b} = 0\). 2. This implies \((2\vec{a} - 3\vec{c})\) is parallel to \(\vec{b}\). So, \(2\vec{a} - 3\vec{c} = \lambda \vec{b}\).

Step 3: Detailed Explanation:
1. Express \(\vec{c}\) in terms of \(\vec{a}, \vec{b},\) and \(\lambda\): \[ 3\vec{c} = 2\vec{a} - \lambda \vec{b} \implies \vec{c} = \frac{2}{3}\vec{a} - \frac{\lambda}{3}\vec{b} \] 2. Use the dot product \(\vec{a} \cdot \vec{c} = 15\): \[ \vec{a} \cdot \left(\frac{2}{3}\vec{a} - \frac{\lambda}{3}\vec{b}\right) = 15 \] \[ \frac{2}{3}|\vec{a}|^2 - \frac{\lambda}{3}(\vec{a} \cdot \vec{b}) = 15 \] - \(|\vec{a}|^2 = 4^2 + (-1)^2 + 3^2 = 16 + 1 + 9 = 26\). - \(\vec{a} \cdot \vec{b} = (4)(10) + (-1)(2) + (3)(-1) = 40 - 2 - 3 = 35\). \[ \frac{2}{3}(26) - \frac{\lambda}{3}(35) = 15 \implies 52 - 35\lambda = 45 \implies 35\lambda = 7 \implies \lambda = \frac{1}{5} \] 3. Find \(\vec{c}\): \[ 3\vec{c} = 2(4\hat{i} - \hat{j} + 3\hat{k}) - \frac{1}{5}(10\hat{i} + 2\hat{j} - \hat{k}) = (8-2)\hat{i} + (-2-2/5)\hat{j} + (6+1/5)\hat{k} \] \[ 3\vec{c} = 6\hat{i} - \frac{12}{5}\hat{j} + \frac{31}{5}\hat{k} \] 4. Find \(\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})\): \[ \frac{1}{3} \left[ (6)(1) + (-12/5)(1) + (31/5)(-3) \right] = \frac{1}{3} \left[ 6 - \frac{12}{5} - \frac{93}{5} \right] = \frac{1}{3} \left[ 6 - \frac{105}{5} \right] = \frac{1}{3} [6 - 21] = -5 \]

Step 4: Final Answer:
The value is -5.