Question

If \(|\vec{A}_1| = 3\), \(|\vec{A}_2| = 4\) and \(|\vec{A}_1 + \vec{A}_2| = 4\), the value of \((2\vec{A}_1 + \vec{A}_2)\cdot(\vec{A}_1 - \vec{A}_2)\) is

• A. \(4.5\)
• B. \(5.5\)
• C. \(6.5\)
• D. \(2.5\)

Hint:

Always use vector magnitude identities to find unknown dot products before expanding expressions.

Answer 1

The Correct Option is C

Step 1: Use vector identity for magnitude.
\[ |\vec{A}_1 + \vec{A}_2|^2 = |\vec{A}_1|^2 + |\vec{A}_2|^2 + 2\vec{A}_1\cdot\vec{A}_2 \] \[ 4^2 = 3^2 + 4^2 + 2\vec{A}_1\cdot\vec{A}_2 \]
Step 2: Calculate dot product \(\vec{A}_1\cdot\vec{A}_2\).
\[ 16 = 9 + 16 + 2\vec{A}_1\cdot\vec{A}_2 \] \[ 2\vec{A}_1\cdot\vec{A}_2 = -9 \Rightarrow \vec{A}_1\cdot\vec{A}_2 = -4.5 \]
Step 3: Expand the given expression.
\[ (2\vec{A}_1 + \vec{A}_2)\cdot(\vec{A}_1 - \vec{A}_2) \] \[ = 2\vec{A}_1\cdot\vec{A}_1 - 2\vec{A}_1\cdot\vec{A}_2 + \vec{A}_2\cdot\vec{A}_1 - \vec{A}_2\cdot\vec{A}_2 \]
Step 4: Substitute values.
\[ = 2(9) - 2(-4.5) + (-4.5) - 16 \] \[ = 18 + 9 - 4.5 - 16 = 6.5 \]
Step 5: Conclusion.
The correct value is \(6.5\).