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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry1 Mark
Question
The position vectors of two points A and B are $\overrightarrow{\text{OA}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$ and $\overrightarrow{\text{OB}}=2\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}},$ respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is ___________.
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Answer

The position vectors of two points A and B are $\overrightarrow{\text{OA}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$ and $\overrightarrow{\text{OB}}=2\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}},$ respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is $2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}.$
Solution:
The position vector are
$\overrightarrow{\text{OA}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$
$\overrightarrow{\text{OB}}=2\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}},$
Let P divide AB, in the ratio such that 2 : 1
$\text{P}=\frac{\text{m}\vec{\text{b}}+\text{n}\vec{\text{a}}}{\text{a+b}}$
$\text{P}=\frac{2(2\vec{\text{i}}-\vec{\text{j}}+2\vec{\text{k}})+1(2\vec{\text{i}}-\vec{\text{j}}-\vec{\text{k}})}{2+1}$
$\text{P}=\frac{6\vec{\text{i}}-3\vec{\text{j}}+3\vec{\text{k}}}{3}$
$=\text{P}=2\vec{\text{i}}-\vec{\text{j}}+\vec{\text{k}}$

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