Question

Find the vector $w$ with the initial point $\left( 9,4 \right)$ and final point $\left( 12,6 \right)$.
A) $\left( 21,10 \right)$
B) $\left( 3,2 \right)$
C) $\left( -21,2 \right)$
D) None of these.

Answer 1

Here, we need to identify the initial point and final point of any vector and then use the formula of finding the vector connecting both points.

Formula used:
The vector $v$ with initial point with the initial point $\left( {{x}_{1}},{{y}_{1}} \right)$ and final point $\left( {{x}_{2}},{{y}_{2}} \right)$ is found by,
$v=\left( {{x}_{2}}-{{x}_{1}},{{y}_{2}}-{{y}_{1}} \right)$

Complete step by step answer:
Consider, the initial and final point which is given in the question,
$\left( 9,4 \right)$ and $\left( 12,6 \right)$ respectively.
The coordinate ${{x}_{1}}$ and ${{y}_{1}}$ come from the initial point and ${{x}_{2}}$ and ${{y}_{2}}$ come from the final point.
We know, that the vector $w$ with initial point with the initial point $\left( {{x}_{1}},{{y}_{1}} \right)$ and final point $\left( {{x}_{2}},{{y}_{2}} \right)$ is found by,
$w=\left( {{x}_{2}}-{{x}_{1}},{{y}_{2}}-{{y}_{1}} \right)$
Putting, ${{x}_{1}}=9$, ${{x}_{2}}=12$, ${{y}_{1}}=4$ and ${{y}_{2}}=6$ on the above equation,

& w=\left( 12-9,6-4 \right) \\\ & w=\left( 3,2 \right) \\\ \end{aligned}$$ Hence, The vector $$w$$ with initial point with the initial point $$\left( {{x}_{1}},{{y}_{1}} \right)$$ and final point $$\left( {{x}_{2}},{{y}_{2}} \right)$$ is obtained as: $$w=\left( 3,2 \right)$$ **Note:** Vector is defined as the quantities which have both magnitude and direction in space. Examples of vectors are: Displacement, velocity, force, acceleration, weight etc. A vector is denoted by a directed line segment (the segment of a line on a plane whose one direction is defined as positive and the opposite direction as negative). Thus, the directed line-segment $$\overline{AB}$$ is a vector. The first letter $$A$$ is called the initial point and the other letter $$B$$ is called the final point of the vector.