First slide

Scalars and Vectors

Question

The angles which a vector $\hat{i} + \hat{j} + \sqrt{2} \hat{k}$ makes with X , Y and Z axes respectively are

Moderate

A

$60^{\circ}, 60^{\circ}, 60^{\circ}$
B

$45^{\circ}, 45^{\circ}, 45^{\circ}$
C

$60^{\circ}, 60^{\circ}, 45^{\circ}$
D

$45^{\circ}, 45^{\circ}, 60^{\circ}$

Solution

$\begin{array}{l} \vec{R} = \hat{i} + \hat{j} + \sqrt{2} \hat{k} \\ Comparing the given vector with \vec{R} = R_{x} \hat{i} + R_{y} \hat{j} + R_{z} \hat{k} \end{array}$
$\begin{array}{l} R_{x} = 1, R_{y} = 1, R_{z} = \sqrt{2} and | \vec{R} | = \sqrt{R_{x}^{2} + R_{y}^{2} + R_{z}^{2}} = 2 \\ \cos α = \frac{R_{x}}{R} = \frac{1}{2} \Rightarrow α = 60^{\circ} \\ \cos β = \frac{R_{y}}{R} = \frac{1}{2} \Rightarrow β = 60^{\circ} \\ \cos γ = \frac{R_{z}}{R} = \frac{1}{\sqrt{2}} \Rightarrow γ = 45^{\circ} \end{array}$

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