How many unit vectors are there for which $$cos$$⁡α$$=12$$ and $$cos$$⁡β$$=12$$,, where α and β are angles made with $$X-axis$$ and $$Y-axis$$, respectively.

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How many unit vectors are there for which $\cos α = \frac{1}{2}$ and $\cos β = \frac{1}{2},$ , where $α$ and $β$ are angles made with X-axis and Y-axis, respectively.

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Solution

$\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$
$\begin{array}{l} or \frac{1}{4} + \frac{1}{4} + \cos^{2} γ = 1 \\ or \cos^{2} γ = 1 - \frac{1}{2} = \frac{1}{2} \\ ∴ \cos γ = \pm \frac{1}{\sqrt{2}} \end{array}$
Two values of cos $γ$ are possible. Hence, two vectors are possible, one corresponding to direction cosines $\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}$ and other corresponding to direction cosines $\frac{1}{2}, \frac{1}{2} - \frac{1}{\sqrt{2}}$

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Scalars and Vectors

Remember concepts with our Masterclasses.

How many unit vectors are there for which cos⁡α=12 and cos⁡β=12,, where α and β are angles made with X-axis and Y-axis, respectively.

cos2⁡α+cos2⁡β+cos2⁡γ=1 or 14+14+cos2⁡γ=1 or cos2⁡γ=1−12=12∴ cos⁡γ=±12Two values of cos γ are possible. Hence, two vectors are possible, one corresponding to direction cosines 12,12,12 and other corresponding to direction cosines 12,12−12

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How many unit vectors are there for which $\cos α = \frac{1}{2}$ and $\cos β = \frac{1}{2},$ , where $α$ and $β$ are angles made with X-axis and Y-axis, respectively.