Question

The angle between the vectors $\overrightarrow {A} = \hat{i} + \hat{j}$ and $\overrightarrow {B} = \hat{i} + \hat{j} + c\hat {k}$ is 30o.Find the unknown $c$ .

• A. zero
• B. 1
• C. $\pm \sqrt{\frac{2}{3}}$
• D. $\pm \frac{1}{2}$

Answer 1

Answer: (C)

$\pm \sqrt{\frac{2}{3}}$

Answer 2

$A = i + j, B = i + j + c\hat{k}$
$A\cdot B = AB\,cos\,\theta$
$( i + j) \cdot (i + j + ck) = \sqrt{2} \times \sqrt{2 + c^2} \times \frac{\sqrt{3}}{2}$
$(\because \theta = 30^{\circ})$
$2 = \sqrt{2} \sqrt{2+ c^2} \times \frac{\sqrt{3}}{2}$
$4 = \sqrt{12 + 6c^2}$
or $16 = 12 + 6c^2$
$c^2 = \frac{4}{6} = \frac{2}{3}$
$c = \pm \sqrt{\frac{2}{3}}$