Question

Let \( \vec{a} + \vec{b} = \lambda \hat{i} + 16\hat{j} - 18\hat{k} \) and \( \vec{a} - \vec{b} = 2\hat{i} + 8\hat{j} + \lambda \hat{k} \). If \( \vec{a} + \vec{b} \) is perpendicular to \( \vec{a} - \vec{b} \), then \( |\vec{a}| = \)

• A. \(5\sqrt{13} \)
• B. \( \sqrt{174} \)
• C. \( \sqrt{184} \)
• D. \( 13\sqrt{5} \)
• E. \( \sqrt{194} \)

Hint:

Use \( 2\vec{a} = (\vec{a}+\vec{b}) + (\vec{a}-\vec{b}) \) to quickly extract vectors.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The dot product of two perpendicular vectors is zero. We will use this property to find the unknown scalar \(\lambda\). After finding \(\lambda\), we can determine the vector \(\vec{a}\) and then calculate its magnitude.
Step 2: Key Formula or Approach:
1. If \(\vec{u} \perp \vec{v}\), then \(\vec{u} \cdot \vec{v} = 0\). 2. Given vectors \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\), we can find \(\vec{a}\) by adding them: \((\vec{a} + \vec{b}) + (\vec{a} - \vec{b}) = 2\vec{a}\). 3. The magnitude of a vector \(\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}\) is \(|\vec{v}| = \sqrt{x^2 + y^2 + z^2}\).
Step 3: Detailed Explanation:
1. Find \(\lambda\). Since \(\vec{a} + \vec{b}\) is perpendicular to \(\vec{a} - \vec{b}\), their dot product is 0. \[ (\lambda \hat{i} + 16\hat{j} - 18\hat{k}) \cdot (2\hat{i} + 8\hat{j} + \lambda \hat{k}) = 0 \] \[ (\lambda)(2) + (16)(8) + (-18)(\lambda) = 0 \] \[ 2\lambda + 128 - 18\lambda = 0 \] \[ 128 - 16\lambda = 0 \] \[ 16\lambda = 128 \] \[ \lambda = \frac{128}{16} = 8 \] 2. Determine the vectors. Now we can write the full expressions for the sum and difference vectors: \[ \vec{a} + \vec{b} = 8\hat{i} + 16\hat{j} - 18\hat{k} \] \[ \vec{a} - \vec{b} = 2\hat{i} + 8\hat{j} + 8\hat{k} \] 3. Find \(\vec{a}\). Add the two vector equations to eliminate \(\vec{b}\): \[ 2\vec{a} = (8\hat{i} + 16\hat{j} - 18\hat{k}) + (2\hat{i} + 8\hat{j} + 8\hat{k}) \] \[ 2\vec{a} = (8+2)\hat{i} + (16+8)\hat{j} + (-18+8)\hat{k} \] \[ 2\vec{a} = 10\hat{i} + 24\hat{j} - 10\hat{k} \] Divide by 2 to get \(\vec{a}\): \[ \vec{a} = 5\hat{i} + 12\hat{j} - 5\hat{k} \] 4. Calculate \(|\vec{a}|\). \[ |\vec{a}| = \sqrt{5^2 + 12^2 + (-5)^2} \] \[ |\vec{a}| = \sqrt{25 + 144 + 25} = \sqrt{194} \] Step 4: Final Answer:
The magnitude of \(\vec{a}\) is \(\sqrt{194}\).