Question

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( \vec{a}\cdot\vec{b} = |\vec{a}\times\vec{b}| \) then the angle between \( \vec{a} \) and \( \vec{b} \) is

• A. \( \frac{\pi}{4} \)
• B. \( -7 \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)

Hint:

Whenever dot product equals magnitude of cross product, directly use \( \cos\theta = \sin\theta \Rightarrow \tan\theta = 1 \).

Answer 1

The Correct Option is A

Step 1: Write dot product formula.
\[ \vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta \]

Step 2: Write cross product magnitude.
\[ |\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta \]

Step 3: Use given condition.
\[ |\vec{a}||\vec{b}|\cos\theta = |\vec{a}||\vec{b}|\sin\theta \]

Step 4: Cancel common terms.
\[ \cos\theta = \sin\theta \]

Step 5: Solve equation.
\[ \tan\theta = 1 \]

Step 6: Find angle.
\[ \theta = \frac{\pi}{4} \]

Step 7: Final conclusion.
\[ \boxed{\frac{\pi}{4}} \]