First slide

Scalars and Vectors

Question

For any two vectors $\vec{A}$ and $\vec{B}$ , if $\bar{A} . \bar{B} = | \bar{A} \times \bar{B} |$ , the magnitude of $\bar{C} = \bar{A} + \bar{B}$ is

Moderate

A

$\sqrt{A^{2} + B^{2}}$
B

$A + B$
C

$\sqrt{A^{2} + B^{2} + \frac{A B}{\sqrt{2}}}$
D

$\sqrt{A^{2} + B^{2} + \sqrt{2} A B}$

Solution

Given $\bar{A} . \bar{B} = | \bar{A} \times \bar{B} | \Rightarrow A B \cos θ = A B \sin θ$
Or $\tan θ = 1 \Rightarrow θ = 45^{0}$

As $\bar{C} = \bar{A} + \bar{B}$

$| \bar{C} | = \sqrt{A^{2} + B^{2} + 2 A B \cos θ} = \sqrt{A^{2} + B^{2} + 2 A B \cos 45^{0}}$

$= \sqrt{A^{2} + B^{2} + \sqrt{2} . A B}$ .

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