Question

Find the dot product of the vectors: \[ (4\hat{i} + 3\hat{j} + 3\hat{k}) \cdot (6\hat{i} - 4\hat{j} + \hat{k}) \]

• A. 22
• B. 15
• C. 21
• D. 18

Hint:

To calculate the dot product, simply multiply the corresponding components of the two vectors and sum the results.

Answer 1

The Correct Option is B

The dot product of two vectors \( \vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) is given by: \[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \] Here, we have: \[ \vec{A} = 4\hat{i} + 3\hat{j} + 3\hat{k}, \quad \vec{B} = 6\hat{i} - 4\hat{j} + \hat{k} \] Now, compute the dot product: \[ \vec{A} \cdot \vec{B} = (4)(6) + (3)(-4) + (3)(1) \] \[ \vec{A} \cdot \vec{B} = 24 - 12 + 3 = 15 \] Thus, the correct answer is: \[ \boxed{15} \]