Question

Find the angle between non-zero vectors \( \mathbf{a} \) and \( \mathbf{b} \) if their dot product \( \mathbf{a}\cdot\mathbf{b} = 0 \).

• A. \(0^\circ\)
• B. \(45^\circ\)
• C. \(90^\circ\)
• D. \(180^\circ\)

Hint:

If the dot product of two non-zero vectors is zero, the vectors are orthogonal (perpendicular), meaning the angle between them is \(90^\circ\).

Answer 1

The Correct Option is C

Concept: The dot product of two vectors is given by \[ \mathbf{a}\cdot\mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\cos\theta \] where \(|\mathbf{a}|\) and \(|\mathbf{b}|\) are the magnitudes of the vectors and \(\theta\) is the angle between them.

Step 1: Use the given condition. \[ \mathbf{a}\cdot\mathbf{b} = 0 \] Substitute into the formula: \[ |\mathbf{a}|\,|\mathbf{b}|\cos\theta = 0 \]

Step 2: Since the vectors are non-zero, their magnitudes are not zero. \[ |\mathbf{a}| \neq 0, \quad |\mathbf{b}| \neq 0 \] Therefore, \[ \cos\theta = 0 \]

Step 3: Find the angle. \[ \theta = 90^\circ \] Thus, the vectors are perpendicular.