If \left | \vec{a} \right |=2,\left | \vec{b} \right |=3\: and\: \left | 2\; \vec{a}-\vec{b} \right | = 5\: then \: \left | 2\; \vec{a} +\vec{b}\right | equals:

  • Option 1)

    17

  • Option 2)

    7

  • Option 3)

    5

  • Option 4)

    1

 

Answers (1)

As we learnt in

Scalar Product of two vectors (dot product) -

\vec{a}\vec{b}=\left | a \right |\left | b \right |Cos\theta

- wherein

\Theta is the angle between the vectors\vec{a}\: and\:\vec{b}

 

 \left | 2\vec{a} \right -\vec{b}|^{2}+\left | 2\vec{a} \right +\vec{b}|^{2}

=\left | 4\vec{a} \right |^{2}+\left | \vec{b} \right |^{2}-4\vec{a}.\vec{b}+\left | 4\vec{a} \right |^{2}+\left | \vec{b} \right |^{2}+4\vec{a}.\vec{b} =8\left | \vec{a} \right |^{2}+2\left | \vec{b} \right |^{2}

Thus 

5^{2}+\left | 2\vec{a} \right +\vec{b}|^{2}=8\times 4+2\times 9

\left | 2\vec{a} \right +\vec{b}|^{2}=32+18-25

\left | 2\vec{a} \right +\vec{b}|=5


Option 1)

17

This solution is incorrect 

Option 2)

7

This solution is incorrect 

Option 3)

5

This solution is correct 

Option 4)

1

This solution is incorrect 

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