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If vec(P).vec(Q)= PQ, then angle between...

If `vec(P).vec(Q)= PQ`, then angle between `vec(P)` and `vec(Q)` is

A

`0^@`

B

`30^@`

C

`45^@`

D

`60^@`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the angle between the vectors \(\vec{P}\) and \(\vec{Q}\) given that \(\vec{P} \cdot \vec{Q} = PQ\). ### Step-by-Step Solution: 1. **Understanding the Dot Product**: The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by the formula: \[ \vec{A} \cdot \vec{B} = |A| |B| \cos(\theta) \] where \(|A|\) and \(|B|\) are the magnitudes of the vectors, and \(\theta\) is the angle between them. 2. **Applying the Given Information**: According to the problem, we have: \[ \vec{P} \cdot \vec{Q} = PQ \] Here, \(P\) and \(Q\) represent the magnitudes of the vectors \(\vec{P}\) and \(\vec{Q}\) respectively. 3. **Using the Dot Product Formula**: From the dot product formula, we can also express the dot product of \(\vec{P}\) and \(\vec{Q}\) as: \[ \vec{P} \cdot \vec{Q} = |\vec{P}| |\vec{Q}| \cos(\theta) = PQ \cos(\theta) \] 4. **Setting the Equations Equal**: Since both expressions equal \(\vec{P} \cdot \vec{Q}\), we can set them equal to each other: \[ PQ \cos(\theta) = PQ \] 5. **Dividing Both Sides by \(PQ\)**: Assuming \(P\) and \(Q\) are not zero, we can divide both sides of the equation by \(PQ\): \[ \cos(\theta) = 1 \] 6. **Finding the Angle**: The cosine of an angle is equal to 1 when the angle is \(0\) degrees. Therefore: \[ \theta = 0^\circ \] ### Final Answer: The angle between \(\vec{P}\) and \(\vec{Q}\) is \(0^\circ\). ---
To solve the problem, we need to find the angle between the vectors \(\vec{P}\) and \(\vec{Q}\) given that \(\vec{P} \cdot \vec{Q} = PQ\). ### Step-by-Step Solution: 1. **Understanding the Dot Product**: The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by the formula: \[ \vec{A} \cdot \vec{B} = |A| |B| \cos(\theta) ...
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