If a vector $$r=xi^+yj^+zk^$$, makes angle π$$3$$,π$$3$$ and πn with $$X-axis$$, $$Y-axis$$ and $$Z-axis$$ respectively. The value of n is $$_______$$.

$$cos2$$⁡α$$+cos2$$⁡β$$+cos2$$⁡γ$$=1$$ or $$cos2$$⁡$$60$$∘$$+cos2$$⁡$$60$$∘$$+cos2$$⁡γ$$=1$$ or $$14+14+cos2$$⁡γ$$=1$$ or $$cos2$$⁡γ$$=1$$−$$12=12$$ or $$cos$$⁡γ$$=12$$⇒γ$$=$$π$$4$$ ∴ $$n=4$$

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If a vector $r = x \hat{i} + y \hat{j} + z \hat{k}$ , makes angle $\frac{π}{3}, \frac{π}{3} and \frac{π}{n}$ with X-axis, Y-axis and Z-axis respectively. The value of n is _______.

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Solution

$\begin{array}{l} \cos^{2} α + \cos^{2} β + \cos^{2} γ = 1 \\ or \cos^{2} 60^{\circ} + \cos^{2} 60^{\circ} + \cos^{2} γ = 1 \\ or \frac{1}{4} + \frac{1}{4} + \cos^{2} γ = 1 \\ or \cos^{2} γ = 1 - \frac{1}{2} = \frac{1}{2} \\ or \cos γ = \frac{1}{\sqrt{2}} \Rightarrow γ = \frac{π}{4} ∴ n = 4 \end{array}$

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Scalars and Vectors

Remember concepts with our Masterclasses.

If a vector r=xi^+yj^+zk^, makes angle π3,π3 and πn with X-axis, Y-axis and Z-axis respectively. The value of n is _______.

cos2⁡α+cos2⁡β+cos2⁡γ=1 or cos2⁡60∘+cos2⁡60∘+cos2⁡γ=1 or 14+14+cos2⁡γ=1 or cos2⁡γ=1−12=12 or cos⁡γ=12⇒γ=π4 ∴ n=4

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If a vector $r = x \hat{i} + y \hat{j} + z \hat{k}$ , makes angle $\frac{π}{3}, \frac{π}{3} and \frac{π}{n}$ with X-axis, Y-axis and Z-axis respectively. The value of n is _______.