Question

If \[ |\vec{a}|=13,\qquad |\vec{b}|=5 \] and \[ \vec{a}\cdot\vec{b}=60, \] then \[ |\vec{a}\times\vec{b}|= \]

• A. \(15\)
• B. \(20\)
• C. \(30\)
• D. \(25\)

Hint:

Remember the identity \[ |\vec{a}\times\vec{b}|^2 = |\vec{a}|^2|\vec{b}|^2-(\vec{a}\cdot\vec{b})^2 \] which is very useful for finding cross products when magnitudes and dot products are known.

Answer 1

The Correct Option is D

Step 1: Use the identity relating dot and cross products.
For any two vectors, \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Substituting the given values, \[ |\vec{a}\times\vec{b}|^2+60^2 = 13^2\times 5^2 \] \[ |\vec{a}\times\vec{b}|^2+3600 = 169\times 25 \] \[ |\vec{a}\times\vec{b}|^2+3600 = 4225 \]

Step 2: Calculate the magnitude of the cross product.
\[ |\vec{a}\times\vec{b}|^2 = 4225-3600 \] \[ |\vec{a}\times\vec{b}|^2 = 625 \] \[ |\vec{a}\times\vec{b}| = 25 \]

Step 3: Final conclusion.
\[ \boxed{25} \]