Question

If \( |\vec{a}|=13, |\vec{b}|=5 \) and \( \vec{a}\cdot\vec{b}=30 \), then \( |\vec{a}\times\vec{b}| \) is

• A. \( 30 \)
• B. \( \frac{30}{25}\sqrt{233} \)
• C. \( \frac{30}{33}\sqrt{193} \)
• D. \( \frac{65}{23}\sqrt{493} \)
• E. \( \frac{65}{13}\sqrt{133} \)

Hint:

Use identity connecting dot and cross product.

Answer 1

Answer: (E)

Concept: \[ |\vec{a}\times\vec{b}| = \sqrt{|\vec{a}|^2|\vec{b}|^2 - (\vec{a}\cdot\vec{b})^2} \]

Step 1: Substitute values.
\[ = \sqrt{(13^2)(5^2) - 30^2} \]

Step 2: Compute squares.
\[ = \sqrt{169 \cdot 25 - 900} \]

Step 3: Simplify.
\[ = \sqrt{4225 - 900} = \sqrt{3325} \]

Step 4: Factor: \[ = \sqrt{25 \cdot 133} = 5\sqrt{133} \]

Step 5: Final: \[ \frac{65}{13}\sqrt{133} \]