If \( \mathbf{a} = 3\hat{i} + 4\hat{j} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} \), find \( \mathbf{a} \cdot \mathbf{b} \) (the dot product).
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2
- 4
- 10
- 12
The Correct Option is A
Solution and Explanation
We are given the following vectors:
\( \mathbf{a} = 3\hat{i} + 4\hat{j} \)
\( \mathbf{b} = 2\hat{i} - \hat{j} \)
Step 1: Recall the formula for the dot product of two vectors:
The dot product of two vectors \( \mathbf{a} = a_1 \hat{i} + a_2 \hat{j} \) and \( \mathbf{b} = b_1 \hat{i} + b_2 \hat{j} \) is given by:
\[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \]
Step 2: Substitute the components of the vectors into the formula:
For \( \mathbf{a} = 3\hat{i} + 4\hat{j} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} \), we have:
\( a_1 = 3, \, a_2 = 4, \, b_1 = 2, \, b_2 = -1 \)
Substitute these values into the dot product formula:
\[ \mathbf{a} \cdot \mathbf{b} = (3)(2) + (4)(-1) = 6 - 4 = 2 \]
Conclusion:
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is \( 2 \).
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