KCET · Maths · Vector Algebra
If \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=-|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|\), then the angle between \(\overrightarrow{\mathbf{a}}\) and \(\vec{b}\) is
- A \(45^{\circ}\)
- B \(180^{\circ}\)
- C \(90^{\circ}\)
- D \(60^{\circ}\)
Answer & Solution
Correct Answer
(B) \(180^{\circ}\)
Step-by-step Solution
Detailed explanation
Given, \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=-|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|\)
\(\Rightarrow \quad|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}| \cos \theta=-|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|\)
\(\Rightarrow \quad \cos \theta=-1\)
\(\Rightarrow \quad \theta=180^{\circ}\)
\(\Rightarrow \quad|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}| \cos \theta=-|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|\)
\(\Rightarrow \quad \cos \theta=-1\)
\(\Rightarrow \quad \theta=180^{\circ}\)
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