Question

If \(|\overrightarrow{a}|=3,|\overrightarrow{b}|=4\) and \(|\overrightarrow{a}-\overrightarrow{b}|=\sqrt7\), then \(\overrightarrow{a}\cdot\overrightarrow{b}\) is equal to

• A. 7
• B. 8
• C. 9
• D. 10
• E. 12

Answer 1

The Correct Option is C

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.