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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors2 Marks
Question
Write the position vector of a point dividing the line segment joining points A and B with position vectors $\vec{\text{a}}\text{ and }\vec{\text{b}}$ externally in the ratio 1 : 4, where $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}$ and $\vec{\text{b}}=-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$.
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Answer

The position vectors of A and B are
$\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}$
$\vec{\text{b}}=-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$
Let C divides AB in the ratio such that AB : CB = 1 : 4
Position vector of $\text{C}=\frac{1\big(-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big)-4\big(2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}\big)}{1-4}$
$=\frac{-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}-8\hat{\text{i}}-12\hat{\text{j}}-16\hat{\text{k}}}{-3}$
$=\frac{-9\hat{\text{i}}-11\hat{\text{j}}-15\hat{\text{k}}}{-3}$
$=3\hat{\text{i}}+\frac{11\hat{\text{j}}}3+5\hat{\text{k}}$

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