First slide

Scalars and Vectors

Question

The angle between vectors $A = 3 \hat{i} + 4 \hat{j} + 5 \hat{k}$ and $B = 6 \hat{i} + 8 \hat{j} + 10 \hat{k}$ is

Moderate

A

Zero
B

$45^{0}$
C

$90^{0}$
D

$180^{0}$

Solution

We know that $A . B = A B \cos θ$                                (i)
                                $A . B = (3 \hat{i} + 4 j + 5 \hat{k}) . (6 \hat{i} + 8 \hat{j} + 10 \hat{k})$                   (ii)

                                $= 3 \times 6 + 4 \times 8 + 5 \times 10 = 100$

The magnitudes of A and B are

                $A = \sqrt{{(3)}^{2} + {(4)}^{2} + {(5)}^{2}} = \sqrt{50}$                                                (iii)

And        $B = \sqrt{{(6)}^{2} + {(8)}^{2} + {(10)}^{2}} = \sqrt{200}$                                           (iv)

Using (ii), (iii) and (iv) in (i), we have

                $100 = \sqrt{50} \times \sqrt{200} \cos θ = 100 \cos θ$

Or           $\cos θ = 1$ or $θ = zero$ .

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