Answer:
[tex]P=\left(\dfrac{13}{3},-\dfrac{1}{3}\right)[/tex]
[tex]|\overrightarrow{MN}|=4\sqrt{5}[/tex]
Step-by-step explanation:
Part (a)
[tex]x_P=\dfrac13(x_N-x_M)+x_M=\dfrac13(7-3)+3=\dfrac{13}{3}[/tex]
[tex]y_P=\dfrac13(y_N-y_M)+y_M=\dfrac13(5-(-3))+(-3)=-\dfrac13[/tex]
[tex]\implies P=\left(\dfrac{13}{3},-\dfrac{1}{3}\right)[/tex]
Part (b)
The magnitude of MN is the distance between points M and N.
Using the distance between two points formula, where
[tex](x_1,y_1)=(3,-3)[/tex] and [tex](x_2,y_2)=(7,5)[/tex]
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]\implies |\overrightarrow{MN}|=\sqrt{(7-3)^2+(5-(-3))^2}[/tex]
[tex]\implies |\overrightarrow{MN}|=4\sqrt{5}[/tex]
