Question

Let $\vec{a} = \hat{i}+\hat{j}+2\hat{k}$ and $\vec{b} = -\hat{i}+2\hat{j}+3\hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times ((\vec{a} \times ((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal to :

• A. $5(30\hat{i} - 5\hat{j} + 7\hat{k})$
• B. $7(30\hat{i} - 5\hat{j} + 7\hat{k})$
• C. $5(34\hat{i} - 5\hat{j} + 3\hat{k})$
• D. $7(34\hat{i} - 5\hat{j} + 3\hat{k})$

Hint:

For complex vector expressions, work systematically from the innermost parentheses outward. Be meticulous with the determinant calculations for cross products to avoid sign errors, which are common mistakes.

Answer 1

The Correct Option is D

Let's evaluate the expression step by step, from the inside out. 
First, calculate $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$. 
$\vec{a} = \hat{i}+\hat{j}+2\hat{k}$ 
$\vec{b} = -\hat{i}+2\hat{j}+3\hat{k}$ 
$\vec{a}+\vec{b} = (1-1)\hat{i} + (1+2)\hat{j} + (2+3)\hat{k} = 3\hat{j} + 5\hat{k}$. 
$\vec{a}-\vec{b} = (1-(-1))\hat{i} + (1-2)\hat{j} + (2-3)\hat{k} = 2\hat{i} - \hat{j} - \hat{k}$. 
Next, calculate the innermost cross product, let $\vec{v}_1 = (\vec{a}-\vec{b}) \times \vec{b}$. 
$\vec{v}_1 = (2\hat{i} - \hat{j} - \hat{k}) \times (-\hat{i}+2\hat{j}+3\hat{k})$