Question

Find the scalar components of the vector \[\overrightarrow{AB}\] with initial point \[A\left( 2,1 \right)\] and terminal point \[B(-5,7)\].

Answer 1

Hint: In this question, we are given with two coordinate points. The initial point is given by \[A\left( 2,1 \right)\] and the terminal point is \[B(-5,7)\].now in order to find the scalar components of the vector \[\overrightarrow{AB}\], we will have to find the coordinates of the point by subtracting the coordinates of point \[A\left( 2,1 \right)\] from the corresponding coordinates of point \[B(-5,7)\]. Then the \[x\] -coordinate of the resultant point as well as the \[y\] -coordinate of the resultant point will become the scalar components of the vector \[\overrightarrow{AB}\].

Complete step by step answer:
We are given the initial and the terminal point of a vector \[\overrightarrow{AB}\].
The initial point is given by \[A\left( 2,1 \right)\] and the terminal point is \[B(-5,7)\] of the vector \[\overrightarrow{AB}\] as shown in the figure given below.

Now let us denote the initial point of the vector \[\overrightarrow{AB}\] by \[\left( {{x}_{1}},{{y}_{1}} \right)\].
That is, we have
\[A\left( 2,1 \right)=\left( {{x}_{1}},{{y}_{1}} \right)\]
Also let us denote the terminal point of the vector \[\overrightarrow{AB}\] by \[\left( {{x}_{2}},{{y}_{2}} \right)\].
That is, we have
\[B(-5,7)=\left( {{x}_{2}},{{y}_{2}} \right)\]
Now we will have to find the coordinates of the point say \[C\] by subtracting the coordinates of point \[B(-5,7)\] from the corresponding coordinates of point \[A\left( 2,1 \right)\].
That is coordinates of the point say \[C\] is given by \[\left( {{x}_{2}},{{y}_{2}} \right)-\left( {{x}_{1}},{{y}_{1}} \right)\], where \[\left( {{x}_{1}},{{y}_{1}} \right)-\left( {{x}_{2}},{{y}_{2}} \right)\] can be calculated by subtracting the corresponding elements.
We will first calculate the value of \[{{x}_{2}}-{{x}_{1}}\].
Since \[{{x}_{1}}=2\] and \[{{x}_{2}}=-5\], thus we have
\[\begin{align}
  & {{x}_{2}}-{{x}_{1}}=-5-\left( 2 \right) \\
 & =-7
\end{align}\]
We will now calculate the value of \[{{y}_{2}}-{{y}_{1}}\].
Since \[{{y}_{1}}=1\] and \[{{y}_{2}}=7\], thus we have
\[\begin{align}
  & {{y}_{2}}-{{y}_{1}}=7-\left( 1 \right) \\
 & =6
\end{align}\]
Hence the coordinates of the point \[C\] which is given by \[\left( {{x}_{2}}-{{x}_{1}},{{y}_{2}}-{{y}_{1}} \right)\] is evaluated as
\[\left( {{x}_{2}}-{{x}_{1}},{{y}_{2}}-{{y}_{1}} \right)=\left( -7,6 \right)\]
Now since the scalar components of the vector \[\overrightarrow{AB}\] are given by \[{{x}_{2}}-{{x}_{1}}\] and \[{{y}_{2}}-{{y}_{1}}\].

Therefore we get that scalar component of the vector \[\overrightarrow{AB}\] are \[-7\] and \[6\].

Note: In this problem, we can simply evaluate the scalar coordinates of the vector \[\overrightarrow{AB}\] by subtracting the coordinates of the point \[A\left( 2,1 \right)\] from the coordinates of the point \[B(-5,7)\].we have to take care that the coordinates of the initial point of the vector \[\overrightarrow{AB}\] is to be subtracted from the corresponding coordinates of the terminal point of the vector \[\overrightarrow{AB}\].