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

Step 1: Understanding the Question:
The question asks for the angle between two vectors, given that they are non-zero and their scalar (dot) product is zero.
Step 2: Key Formula or Approach:
The geometric definition of the dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) 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 3: Detailed Explanation:
We are given that \( \mathbf{a} \cdot \mathbf{b} = 0 \). Substituting this into the formula:
\[ |\mathbf{a}| |\mathbf{b}| \cos\theta = 0 \] The problem also states that \( \mathbf{a} \) and \( \mathbf{b} \) are non-zero vectors. This means their magnitudes are not zero:
\[ |\mathbf{a}| \neq 0 \quad \text{and} \quad |\mathbf{b}| \neq 0 \] For the product of three terms to be zero, at least one of them must be zero. Since we know \(|\mathbf{a}|\) and \(|\mathbf{b}|\) are not zero, it must be that \(\cos\theta\) is zero.
\[ \cos\theta = 0 \] The angle \(\theta\) for which \(\cos\theta = 0\) (in the range \(0^\circ \leq \theta \leq 180^\circ\)) is \(90^\circ\). This means the vectors are perpendicular or orthogonal.
Step 4: Final Answer:
The angle between the vectors is \(90^\circ\).