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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors5 Marks
Question
If the vectors $\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}$ and $\vec{\text{b}}=-6\hat{\text{i}}+\text{m}\hat{\text{j}}$ are collinear, find tghe value of m.
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Answer

Here, it is given that vectors $\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}$ and $\vec{\text{b}}=-6\hat{\text{i}}+\text{m}\hat{\text{j}}$ are collinear. So, $\text{a}=\lambda\text{b}$, for a scalar $\lambda$$2\hat{\text{i}}-3\hat{\text{j}}=\lambda\big(-6\hat{\text{i}}+\text{m}\hat{\text{j}}\big)$
$2\hat{\text{i}}-3\hat{\text{j}}=-6\lambda\hat{\text{i}}+\text{m}\lambda\hat{\text{j}}\big)$ Comparing the coefficients of LHS and RHS, $2=-6\lambda$ $\lambda=\frac{2}{-6}$ $\lambda=\frac{-1}3\ \dots(\text{i})$ $-3=\lambda\text{m}$ $\lambda=\frac{-3}{\text{m}}\ \dots(\text{ii})$ From (i) and (ii), $\frac{-1}3=\frac{-3}{\text{m}}$ $\text{m}=3\times3$ $=9$ $\therefore\ \text{m}=9$

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