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

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

& {{x}_{2}}-{{x}_{1}}=-5-\left( 2 \right) \\\ & =-7 \end{aligned}$$ We will now calculate the value of $${{y}_{2}}-{{y}_{1}}$$. Since $${{y}_{1}}=1$$ and $${{y}_{2}}=7$$, thus we have $$\begin{aligned} & {{y}_{2}}-{{y}_{1}}=7-\left( 1 \right) \\\ & =6 \end{aligned}$$ Hence the coordinates of the point $$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}$$.