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Scalar Or Dot Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product2 Marks
Question
Write the value of p for which $\vec{\text{a}}=3\hat{\text{i}}+2\hat{\text{j}}+9\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+\text{p}\hat{\text{j}}+3\hat{\text{k}}$ are parallel vectors.
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Answer

We have$\vec{\text{a}}=3\hat{\text{i}}+2\hat{\text{j}}+9\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+\text{p}\hat{\text{j}}+3\hat{\text{k}}$
Given that $\vec{\text{a}}$ and $\vec{\text{b}}$ are parallel.
$\Rightarrow\vec{\text{a}}=\text{t}\vec{\text{b}}$ for some t.
$\Rightarrow3\hat{\text{i}}+2\hat{\text{j}}+9\hat{\text{k}}=\text{t}\big(\hat{\text{i}}+\text{p}\hat{\text{j}}+3\hat{\text{k}}\big)$
$\Rightarrow3\hat{\text{i}}+2\hat{\text{j}}+9\hat{\text{k}}=\text{t}\hat{\text{i}}+\text{pt}\hat{\text{j}}+3\text{t}\hat{\text{k}}$
Comparing both sides, we get
$3=\text{t,}2=\text{pt}$ and $9=3\text{t}$
$\Rightarrow\text{t}=3$ and $\text{pt}=2$
$\Rightarrow3\text{t}=2$
$\therefore\text{t}=\frac{2}{3}$

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