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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA5 Marks
Question
The position vectors of points A, B and C are $\lambda\hat{\text{i}}+3\hat{\text{j}},12\hat{\text{i}}+\mu\hat{\text{j}}\text{ and }11\hat{\text{i}}-3\hat{\text{j}}$ respectively. If C divides the line segment joining A and B in the ratio 3:1, find the value of $\lambda\text{ and }\mu$
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Answer

The position vectors of points A, B and C are $\lambda\hat{\text{i}}+3\hat{\text{j}},12\hat{\text{i}}+\mu\hat{\text{j}}\text{ and }11\hat{\text{i}}-3\hat{\text{j}}$, respectively.
It is given that, C divides the line segment joining A and B in the ratio 3 : 1.
$11\hat{\text{i}}-3\hat{\text{j}}=\frac{3\times\big(12\hat{\text{i}}+\mu\hat{\text{j}}\big)+1\times\big(\lambda\hat{\text{i}}+3\hat{\text{j}}\big)}{3+1}$
$\Rightarrow11\hat{\text{i}}-3\hat{\text{j}}=\frac{(36+\lambda)\hat{\text{i}}+(3\mu+3)\hat{\text{j}}}{4}$
$\Rightarrow44\hat{\text{i}}-12\hat{\text{j}}=(36+\lambda)\hat{\text{i}}+(3\mu+3)\hat{\text{j}}$
Equating the corresponding components, we get
$36+\lambda=44$
$\Rightarrow\lambda=44-36=8$
and
$3\mu+3=-12$
$\Rightarrow3\mu=-12-3$
$\Rightarrow\mu=-5$
Thus the values of $\lambda\text{ and }\mu$ are 8 and -5, respectively.

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