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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 their dot product is equal to the product of their magnitudes. ### Step-by-Step Solution: 1. **Understanding the Dot Product**: The dot product of two vectors \(\vec{P}\) and \(\vec{Q}\) is given by the formula: \[ \vec{P} \cdot \vec{Q} = |\vec{P}| |\vec{Q}| \cos \theta \] where \(\theta\) is the angle between the two vectors. 2. **Setting Up the Equation**: According to the question, we have: \[ \vec{P} \cdot \vec{Q} = PQ \] Here, \(PQ\) represents the product of the magnitudes of vectors \(\vec{P}\) and \(\vec{Q}\). Thus, we can rewrite the equation as: \[ |\vec{P}| |\vec{Q}| \cos \theta = |\vec{P}| |\vec{Q}| \] 3. **Dividing Both Sides**: Assuming \(|\vec{P}|\) and \(|\vec{Q}|\) are not zero, we can divide both sides of the equation by \(|\vec{P}| |\vec{Q}|\): \[ \cos \theta = 1 \] 4. **Finding the Angle**: The cosine of an angle is equal to 1 when the angle is: \[ \theta = \cos^{-1}(1) = 0^\circ \] 5. **Conclusion**: Therefore, the angle between the vectors \(\vec{P}\) and \(\vec{Q}\) is: \[ \theta = 0^\circ \]
To solve the problem, we need to find the angle between the vectors \(\vec{P}\) and \(\vec{Q}\) given that their dot product is equal to the product of their magnitudes. ### Step-by-Step Solution: 1. **Understanding the Dot Product**: The dot product of two vectors \(\vec{P}\) and \(\vec{Q}\) is given by the formula: \[ \vec{P} \cdot \vec{Q} = |\vec{P}| |\vec{Q}| \cos \theta ...
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Knowledge Check

  • Given that P=Q=R . If vec(P)+vec(Q)=vec(R) then the angle between vec(P) and vec(R) is theta_(1) . If vec(P)+vec(Q)+vec(R)=vec(0) then the angle between vec(P) and vec(R) is theta_(2) . The relation between theta_(1) and theta_(2) is :-

    A
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    B
    `theta_(1)=theta_(2)/2`
    C
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    D
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  • Given vec(p)*(vec(P)+vec(Q))=P^(2) then the angle between vec(P)andvec(Q) is

    A
    `0^(@)`
    B
    `30^(@)`
    C
    `45^(@)`
    D
    `90^(@)`

  • Three vectors vec(P) , vec(Q) and vec( R) are such that |vec(P)| , |vec(Q )|, |vec(R )| = sqrt(2) |vec(P)| and vec(P) + vec(Q) + vec(R ) = 0 . The angle between vec(P) and vec(Q) , vec(Q) and vec(R ) and vec(P) and vec(R ) are

    A
    `90^(@) , 135^(@) , 135^(@)`
    B
    `90^(@) , 45^(@) , 45^(@)`
    C
    `45^(@) , 90^(@) , 90^(@)`
    D
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