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The resultant vec(P) and vec(Q) is perpe...

The resultant `vec(P) and vec(Q)` is perpendicular to `vec(P)`. What is the angle between `vec(P) and vec(Q)`?

A

`cos^(-1)((P)/(Q))`

B

`cos^(-1)((-P)/(Q))`

C

`sin^(-1)((P)/(Q))`

D

`sin^(-1)((-P)/(Q))`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the angle between the vectors \(\vec{P}\) and \(\vec{Q}\) given that their resultant \(\vec{R} = \vec{P} + \vec{Q}\) is perpendicular to \(\vec{P}\). ### Step-by-Step Solution: 1. **Understanding the Condition**: We know that the resultant vector \(\vec{R}\) is given by: \[ \vec{R} = \vec{P} + \vec{Q} \] Since \(\vec{R}\) is perpendicular to \(\vec{P}\), we can use the dot product property: \[ \vec{R} \cdot \vec{P} = 0 \] 2. **Expanding the Dot Product**: Substitute \(\vec{R}\) into the dot product equation: \[ (\vec{P} + \vec{Q}) \cdot \vec{P} = 0 \] This expands to: \[ \vec{P} \cdot \vec{P} + \vec{Q} \cdot \vec{P} = 0 \] 3. **Using Magnitudes**: Let \(P = |\vec{P}|\) and \(Q = |\vec{Q}|\). The dot product \(\vec{Q} \cdot \vec{P}\) can be expressed as: \[ \vec{Q} \cdot \vec{P} = |\vec{Q}| |\vec{P}| \cos(\theta) \] where \(\theta\) is the angle between \(\vec{P}\) and \(\vec{Q}\). 4. **Substituting Back**: Now, substituting this back into the equation gives: \[ P^2 + QP \cos(\theta) = 0 \] 5. **Rearranging the Equation**: Rearranging the equation leads to: \[ QP \cos(\theta) = -P^2 \] Dividing both sides by \(QP\) (assuming \(Q \neq 0\)): \[ \cos(\theta) = -\frac{P}{Q} \] 6. **Finding the Angle**: To find the angle \(\theta\), we take the inverse cosine: \[ \theta = \cos^{-1}\left(-\frac{P}{Q}\right) \] ### Conclusion: The angle \(\theta\) between the vectors \(\vec{P}\) and \(\vec{Q}\) is given by: \[ \theta = \cos^{-1}\left(-\frac{P}{Q}\right) \]
To solve the problem, we need to find the angle between the vectors \(\vec{P}\) and \(\vec{Q}\) given that their resultant \(\vec{R} = \vec{P} + \vec{Q}\) is perpendicular to \(\vec{P}\). ### Step-by-Step Solution: 1. **Understanding the Condition**: We know that the resultant vector \(\vec{R}\) is given by: \[ \vec{R} = \vec{P} + \vec{Q} ...
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