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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors4 Marks
Question
If $\vec{\text{a}}$ be the position vector whose tip is (5, -3), find the coordinates of a point B such that $\overrightarrow{\text{AB}}=\vec{\text{a}}$, the coordinates of A being (4, -1).
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Answer

Let O be the origin and let P(5, -3) be the tip of the position vector $\vec{\text{a}}$. Then,
$\vec{\text{a}}=\overrightarrow{\text{OP}}=5\hat{\text{i}}-3\hat{\text{j}}$. Let the coordinate of B be (x, y) and A has coordinates (4, -1).
Therefore,
$\overrightarrow{\text{AB}}$ = Position vector of B - Position vector of A
$=\big(\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}\big)-\big(4\hat{\text{i}}-\hat{\text{j}}\big)$
$=(\text{x}-4)\hat{\text{i}}+(\text{y}+1)\hat{\text{j}}$
Now,
$\overrightarrow{\text{AB}}=\vec{\text{a}}$
$\Rightarrow(\text{x}-4)\hat{\text{i}}+(\text{y}+1)\hat{\text{j}}=5\hat{\text{i}}-3\hat{\text{j}}$
$\Rightarrow\text{x}-4=5\text{ and }\text{y}+1=-3$
$\Rightarrow\text{x}=9 \text{ and }\text{y}=-4$
Hence, the coordinates of B are (9, -4).

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