Question

Three vectors are as follows: \[ \mathbf{a} = 3\hat{i} - 10\hat{j} + 7\hat{k}, \mathbf{b} = -9\hat{i} + 6\hat{j} - 47\hat{k}, \mathbf{c} = 11\hat{i} - 17\hat{k} \] The value of \((\mathbf{a} + \mathbf{b}) \cdot \mathbf{c}\) is

• A. 614
• B. 746
• C. 2
• D. 134

Hint:

To calculate the dot product, multiply corresponding components of the vectors and add the results.

Answer 1

The Correct Option is A

Step 1: Compute \(\mathbf{a} + \mathbf{b}\).
\[ \mathbf{a} + \mathbf{b} = (3\hat{i} - 10\hat{j} + 7\hat{k}) + (-9\hat{i} + 6\hat{j} - 47\hat{k}) = (-6\hat{i} - 4\hat{j} - 40\hat{k}) \]

Step 2: Take the dot product with \(\mathbf{c}\).
Now compute the dot product: \[ (\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} = (-6\hat{i} - 4\hat{j} - 40\hat{k}) \cdot (11\hat{i} - 17\hat{k}) \] \[ = (-6)(11) + (-4)(0) + (-40)(-17) = -66 + 0 + 680 = 614 \]

Step 3: Conclusion.
The correct answer is (A) 614, as the dot product yields 614.