Question

If \( \vec{a} = 2\hat{i} - 5\hat{j} + 8\hat{k} \), \( \vec{b} = 7\hat{i} - 5\hat{j} + 3\hat{k} \), and \[ (\vec{2a} - \vec{3b}) \times (\vec{4a} + \vec{b}) = x\hat{i} + y\hat{j} + z\hat{k}, \] then \( x + y + z = \)

• A. \(-1000\)
• B. \(1400\)
• C. \(1000\)
• D. \(-1400\)

Hint:

For vector cross product problems, first simplify both vectors before calculating the determinant. Keep track of signs carefully!

Answer 1

The Correct Option is B

Step 1: Compute \(2\vec{a} - 3\vec{b}\) \[ \begin{aligned} 2\vec{a} &= 4\hat{i} - 10\hat{j} + 16\hat{k}
3\vec{b} &= 21\hat{i} - 15\hat{j} + 9\hat{k}
\Rightarrow 2\vec{a} - 3\vec{b} &= (4 - 21)\hat{i} + (-10 + 15)\hat{j} + (16 - 9)\hat{k}
&= -17\hat{i} + 5\hat{j} + 7\hat{k} \end{aligned} \]
Step 2: Compute \(4\vec{a} + \vec{b}\) \[ 4\vec{a} = 8\hat{i} - 20\hat{j} + 32\hat{k}
\Rightarrow 4\vec{a} + \vec{b} = (8 + 7)\hat{i} + (-20 - 5)\hat{j} + (32 + 3)\hat{k} = 15\hat{i} - 25\hat{j} + 35\hat{k} \]
Step 3: Compute cross product
Let \(\vec{u} = -17\hat{i} + 5\hat{j} + 7\hat{k}\), \(\vec{v} = 15\hat{i} - 25\hat{j} + 35\hat{k}\). Then, \[ \vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
-17 & 5 & 7
15 & -25 & 35 \end{vmatrix} \] \[ = \hat{i}(5 \cdot 35 - 7 \cdot (-25)) - \hat{j}(-17 \cdot 35 - 7 \cdot 15) + \hat{k}(-17 \cdot (-25) - 5 \cdot 15) \] \[ = \hat{i}(175 + 175) - \hat{j}(-595 - 105) + \hat{k}(425 - 75) \] \[ = \hat{i}(350) + \hat{j}(700) + \hat{k}(350) \] So, \( x = 350, y = 700, z = 350 \Rightarrow x + y + z = 1400 \)