If \(|\overrightarrow{a}|=3,|\overrightarrow{b}|=4\) and \(|\overrightarrow{a}-\overrightarrow{b}|=\sqrt7\), then \(\overrightarrow{a}\cdot\overrightarrow{b}\) is equal to
- 7
- 8
- 9
- 10
- 12
The Correct Option is C
Solution and Explanation
Step 1: Use the magnitude identity
The magnitude of the difference of two vectors is given by: \[ |\overrightarrow{a} - \overrightarrow{b}|^2 = |\overrightarrow{a}|^2 + |\overrightarrow{b}|^2 - 2\overrightarrow{a} \cdot \overrightarrow{b} \] Given: \[ |\overrightarrow{a}| = 3, \quad |\overrightarrow{b}| = 4, \quad |\overrightarrow{a} - \overrightarrow{b}| = \sqrt{7} \] Squaring both sides: \[ 7 = 3^2 + 4^2 - 2\overrightarrow{a} \cdot \overrightarrow{b} \]
Step 2: Solve for \( \overrightarrow{a} \cdot \overrightarrow{b} \)
\[ 7 = 9 + 16 - 2\overrightarrow{a} \cdot \overrightarrow{b} \] \[ 7 = 25 - 2\overrightarrow{a} \cdot \overrightarrow{b} \] \[ 2\overrightarrow{a} \cdot \overrightarrow{b} = 18 \] \[ \overrightarrow{a} \cdot \overrightarrow{b} = 9 \]
Final Answer: \( \overrightarrow{a} \cdot \overrightarrow{b} \) is 9.
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