Question

If \(|\vec{u}|=5,|\vec{v}|=13,|\vec{u}\times \vec{v}|=25\), then \(\vec{u}\cdot \vec{v}=...\)

• A. \(15\)
• B. \(\pm \frac{5}{13}\)
• C. \(\pm 60\)
• D. \(\pm 65\)

Hint:

Use Lagrange's identity: |u x v|^2 + (u.v)^2 = |u|^2 |v|^2.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Lagrange's identity: \(|\vec u\times\vec v|^2+(\vec u\cdot\vec v)^2=|\vec u|^2|\vec v|^2\).

Step 2: Substitute
\(625+(\vec u\cdot\vec v)^2=25\times169=4225\).

Step 3: Solve
\((\vec u\cdot\vec v)^2=3600\), so \(\vec u\cdot\vec v=\pm60\).

Step 4: Compare the options
The sign is not fixed by the data, so the answer is \(\pm60\).

Final Answer:
The dot product is \(\pm60\). \[ \boxed{\text{(C) }\pm 60} \]