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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA2 Marks
Question
If a vector makes angles $\alpha,\beta,\gamma$ with OX, OY and OZ respectively. then write the value of $\sin^2\alpha+\sin^2\beta+\sin^2\gamma$.
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Answer

Suppose, a vector $\overrightarrow{\text{OP}}$ makes an angle $\alpha,\beta,\gamma$ with OX, OY and OZ respectively.Then direction consines of the vector are given by $\text{l}=\cos\alpha,\ \text{m}=\cos\beta,\ \text{n}=\cos\gamma$

Consider,

 $\sin^2\alpha+\sin^2\beta+\sin^2\gamma\\=1-\cos^2\alpha+1-\cos^2\beta+1-\cos^2\gamma$

$=3-(\cos^2\alpha+\cos^2\beta+\cos^2\gamma)$

$=3-(\text{l}^2+\text{m}^2+\text{n}^2)$

$=3-1$ $[\because\ \text{l}^2+\text{m}^2+\text{n}^2=1]$

$=2$

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