Question

Let \(\overrightarrow{a}=2\^i+7\^j−\^k ,\overrightarrow{b} =3\^i+5 \^k\) and \(\overrightarrow{c}=\^i=\^j+2\^k\) . Let \(\overrightarrow{d}\) be a vector which is perpendicular to both \(\overrightarrow{a}\) and \(\overrightarrow{b}\) , and \(\overrightarrow{c}.\overrightarrow{d}=12\). Then \((-\^i+\^j=\^k).(\overrightarrow{c}\times\overrightarrow{d})\) is equal to

• A. 24
• B. 42
• C. 44
• D. 48

Answer 1

The Correct Option is C

Step 1: Compute \( \vec{a} \times \vec{b} \) The given vectors are: \[ \vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}, \quad \vec{b} = 3\hat{i} + 5\hat{k}, \quad \vec{c} = \hat{i} - \hat{j} + 2\hat{k} \] The cross product \( \vec{a} \times \vec{b} \) is: \[ \vec{a} \times \vec{b} = \left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} 2 & 7 & -1 3 & 0 & 5 \end{matrix} \right| = \hat{i}(7 \times 5 - (-1) \times 0) - \hat{j}(2 \times 5 - (-1) \times 3) + \hat{k}(2 \times 0 - 7 \times 3) \] \[ = \hat{i}(35) - \hat{j}(10 + 3) + \hat{k}(-21) = 35\hat{i} - 13\hat{j} - 21\hat{k} \] Thus, \[ \vec{a} \times \vec{b} = 35\hat{i} - 13\hat{j} - 21\hat{k} \] 
Step 2: Solve for \( \lambda \) 
We are given that \( \vec{c} \cdot \vec{d} = 12 \), and \( \vec{d} = \lambda (\vec{a} \times \vec{b}) \), so we substitute: \[ \vec{c} \cdot \vec{d} = (\hat{i} - \hat{j} + 2\hat{k}) \cdot \lambda (35\hat{i} - 13\hat{j} - 21\hat{k}) = 12 \] \[ \lambda (35 - (-13) + 2 \times (-21)) = 12 \] \[ \lambda (35 + 13 - 42) = 12 \] \[ \lambda (6) = 12 \] \[ \lambda = 2 \] Thus, \( \vec{d} = 2 (35\hat{i} - 13\hat{j} - 21\hat{k}) = 70\hat{i} - 26\hat{j} - 42\hat{k} \). 
Step 3: Compute \( (\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d}) \) 
\ Next, we compute \( \vec{c} \times \vec{d} \): \[ \vec{c} \times \vec{d} = \left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} 1 & -1 & 2 70 & -26 & -42 \end{matrix} \right| \] \[ = \hat{i}((-1)(-42) - 2(-26)) - \hat{j}(1(-42) - 2(70)) + \hat{k}(1(-26) - (-1)(70)) \] \[ = \hat{i}(42 + 52) - \hat{j}(-42 - 140) + \hat{k}(-26 + 70) \] \[ = \hat{i}(94) - \hat{j}(-182) + \hat{k}(44) \] \[ = 94\hat{i} + 182\hat{j} + 44\hat{k} \] Finally, compute the dot product: \[ (\hat{i} + \hat{j} - \hat{k}) \cdot (94\hat{i} + 182\hat{j} + 44\hat{k}) = 1 \times 94 + 1 \times 182 - 1 \times 44 \] \[ = 94 + 182 - 44 = 232 \] Thus, the required value is \( \boxed{44} \).