Question

If \(|\vec{a}| = 5\), \(|\vec{b}| = 13\), and \(|\vec{a} \times \vec{b}| = 25\), then \(|\vec{a} \cdot \vec{b}|\) is equal to ........

• A. 30
• B. 60
• C. 40
• D. 45

Hint:

For problems involving the dot product and cross product, use the trigonometric identities to relate \(\sin \theta\) and \(\cos \theta\) to solve the equation.

Answer 1

The Correct Option is B

Step 1: Recall the relationship between the dot product and cross product.
We know that: \[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \text{and} \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

Step 2: Use the given values.
From the problem, we have: \[ |\vec{a}| = 5, \; |\vec{b}| = 13, \; |\vec{a} \times \vec{b}| = 25 \] \[ 25 = 5 \times 13 \times \sin \theta $\Rightarrow$ \sin \theta = \frac{25}{65} = \frac{5}{13} \]

Step 3: Find \(\cos \theta\).
Since \(\sin^2 \theta + \cos^2 \theta = 1\), we find: \[ \cos^2 \theta = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} \] \[ \cos \theta = \frac{12}{13} \]

Step 4: Calculate the dot product.
\[ \vec{a} \cdot \vec{b} = 5 \times 13 \times \frac{12}{13} = 60 \]

Step 5: Conclude.
The value of \(|\vec{a} \cdot \vec{b}|\) is 60.

Final Answer: \[ \boxed{60} \]