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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors2 Marks
Question
Find the value of 'p' for which the vectors $3\hat{\text{i}}+2\hat{\text{j}}+9\hat{\text{k}}$ and $\hat{\text{i}}-2\text{p}\hat{\text{j}}+3\hat{\text{k}}$ are parallel.
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Answer

Let $\vec{\text{a}}=3\hat{\text{i}}+2\hat{\text{j}}+9\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}-2\text{p}\hat{\text{j}}+3\hat{\text{k}}$ be the two given vectors.
If $\vec{\text{a}}\text{ and }\vec{\text{b}}$ are parallel, then
$\vec{\text{b}}=\lambda\vec{\text{a}}$ for some scalar $\lambda$
$\therefore\ \hat{\text{i}}-2\text{p}\hat{\text{j}}+3\hat{\text{k}}=\lambda\big(3\hat{\text{i}}-2\hat{\text{j}}+9\hat{\text{k}}\big)$
$\Rightarrow\ \hat{\text{i}}-2\text{p}\hat{\text{j}}+3\hat{\text{k}}=3\lambda\hat{\text{i}}+2\lambda\hat{\text{j}}+9\lambda\hat{\text{k}}$
$\Rightarrow\ 1=\lambda3$ and $-2\text{p}=2\lambda$ $\big($ Equating coefficients of $\hat{\text{i}},\hat{\text{j}},\hat{\text{k}}\big)$
$\Rightarrow\ \text{p}=-\lambda=-\frac{1}3$
Thus, the value of p is $-\frac{1}3$.

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