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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors2 Marks
Question
Write two different vectors having same direction.
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Answer

Let $\vec{\text{p}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{q}}=2\hat{\text{i}}+4\hat{\text{j}}+6\hat{\text{k}}$
Then, direction cosines of $\vec{\text{p}}$ are
$\text{l}=\frac{1}{\sqrt{1^2+2^2+3^2}}=\frac{1}{\sqrt{14}},\text{m}=\frac{2}{\sqrt{1^2+2^2+3^2}}=\frac{2}{\sqrt{14}}$ and $\text{n}=\frac{3}{\sqrt{1^2+2^2+3^2}}=\frac{3}{\sqrt{14}}$
Direction cosines of $\vec{\text{q}}$ are
$\text{l}=\frac{2}{\sqrt{2^2+4^2+6^2}}=\frac{2}{2\sqrt{14}}=\frac{1}{\sqrt{14}},$ $\text{m}=\frac{4}{\sqrt{2^2+4^2+6^2}}=\frac{4}{2\sqrt{14}}=\frac{2}{\sqrt{14}}$ and $\text{n}=\frac{6}{\sqrt{2^2+4^2+6^2}}=\frac{6}{2\sqrt{14}}=\frac{3}{\sqrt{14}}$
The direction cosines of two vectors are same. Hence the two different vectors $\vec{\text{p}},\vec{\text{q}}$ have same directions.

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