Question

Match List-I with List-II. Suppose \(\vec{a}\) and \(\vec{b}\) are vectors. 

List-I:

A. \(\vec{a}\) and \(\vec{b}\) are perpendicular if,
B. If angle between \(\vec{a}\) and \(\vec{b}\) is \(0\), then,
C. If angle between \(\vec{a}\) and \(\vec{b}\) is \(\pi\), then,
D. If \(\vec{a}\) and \(\vec{b}\) are unit vectors, then.

List-II:

I. \(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\)
II. \(\vec{a}\cdot\vec{b} = -|\vec{a}||\vec{b}|\)
III. \(\vec{a}\cdot\vec{b} = \cos\theta\)
IV. \(\vec{a}\cdot\vec{b} = 0\)

• A. A-IV, B-I, C-II, D-III
• B. A-IV, B-II, C-I, D-III
• C. A-I, B-IV, C-II, D-III
• D. A-I, B-II, C-IV, D-III

Hint:

Always start vector angle questions with \(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\).

Answer 1

The Correct Option is A

Concept:
Dot product of two vectors is: \[ \vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta \]

Step 1: Perpendicular vectors.
If vectors are perpendicular, then: \[ \theta=90^\circ \] \[ \cos 90^\circ=0 \] So, \[ \vec{a}\cdot\vec{b}=0 \] Thus, \[ A\rightarrow IV \]

Step 2: Angle is \(0\).
If angle is \(0\), then: \[ \cos 0=1 \] \[ \vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}| \] Thus, \[ B\rightarrow I \]

Step 3: Angle is \(\pi\).
If angle is \(\pi\), then: \[ \cos\pi=-1 \] \[ \vec{a}\cdot\vec{b}=-|\vec{a}||\vec{b}| \] Thus, \[ C\rightarrow II \]

Step 4: Unit vectors.
If \(\vec{a}\) and \(\vec{b}\) are unit vectors, then: \[ |\vec{a}|=|\vec{b}|=1 \] So, \[ \vec{a}\cdot\vec{b}=\cos\theta \] Thus, \[ D\rightarrow III \] Therefore, \[ A-IV,\ B-I,\ C-II,\ D-III \] \[ \therefore \text{Correct Answer is (A)} \]