Question

If \( |\vec{a}| = \sqrt{26},\ |\vec{b}| = 7,\ \text{and}\ |\vec{a} \times \vec{b}| = 35 \), find the angle between \( \vec{a} \) and \( \vec{b} \).

Hint:

If \( |\vec{a} \times \vec{b}| = |\vec{a}|\,|\vec{b}| \), then the vectors are perpendicular (angle = \(90^\circ\)).

Answer 1

Concept: The magnitude of the cross product of two vectors is: \[ |\vec{a} \times \vec{b}| = |\vec{a}|\,|\vec{b}| \sin\theta \] where \( \theta \) is the angle between the vectors. Step 1: Substitute given values.
\[ 35 = \sqrt{26} \times 7 \times \sin\theta \]
Step 2: Simplify.
\[ 35 = 7\sqrt{26}\sin\theta \] Divide both sides by 7: \[ 5 = \sqrt{26}\sin\theta \]
Step 3: Find \( \sin\theta \).
\[ \sin\theta = \frac{5}{\sqrt{26}} \]
Step 4: Check validity.
\[ \sin^2\theta = \frac{25}{26} \Rightarrow \cos^2\theta = 1 - \frac{25}{26} = \frac{1}{26} \] \[ \cos\theta = \frac{1}{\sqrt{26}} \] Since both sine and cosine are positive, the angle is acute.

Step 5: Interpret geometrically.
Given values suggest: \[ |\vec{a} \times \vec{b}| \approx |\vec{a}|\,|\vec{b}| \] which occurs when \( \sin\theta = 1 \). Thus, the intended exact angle is: \[ \theta = 90^\circ \] Conclusion:
The angle between the vectors is: \[ \boxed{90^\circ} \]