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

Hint: 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,
\[\begin{align}
  & w=\left( 12-9,6-4 \right) \\
 & w=\left( 3,2 \right) \\
\end{align}\]
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.