How to determine the coordinates of the point M?#A_(((2,-5)));B_(((-3,5)))#;And #vec(BM)=1/5vec(AB)#

Question

Allison Allen · Accepted Answer

# M=M(-4,7)#.

Suppose that, #M=M(x,y)#. #:. vec(BM)=(x,y)-(-3,5)=(x+3,y-5)............(star_1)#. # vec(AB)=(-3,5)-(2,-5)=(-5,10).................(star_2)#. But, Given that, #vec(BM)=1/5vec(AB)#. #:. (x+3,y-5)=1/5(-5,10)...[because, (star_1) & (star_2)]#. #:. (x+3,y-5)=(-1,2)#. #:. x+3=-1 and y-5=2#. #:. x=-4 and y=7#. #:. M(x,y)=M(-4,7)#.

Sadie Allen · Answer

undefined To determine the coordinates of point M, you can use the vector equation for point M, which is given by:
$$ \vec{OM} = \vec{OA} + \frac{1}{5} \vec{AB} $$

Where:
- $\vec{OA}$ represents the position vector of point A,
- $\vec{AB}$ represents the vector from point A to point B.

Given:
- Coordinates of point A: $ A(2,-5) $
- Coordinates of point B: $ B(-3,5) $

You can find $\vec{AB}$ by subtracting the coordinates of A from the coordinates of B.

$$ \vec{AB} = \begin{pmatrix} -3 - 2 \ 5 - (-5) \end{pmatrix} = \begin{pmatrix} -5 \ 10 \end{pmatrix} $$

Then, multiply $\vec{AB}$ by $\frac{1}{5}$ to find $ \frac{1}{5} \vec{AB} $.

$$ \frac{1}{5} \vec{AB} = \frac{1}{5} \begin{pmatrix} -5 \ 10 \end{pmatrix} = \begin{pmatrix} -1 \ 2 \end{pmatrix} $$

Finally, add $\vec{OA}$ to $ \frac{1}{5} \vec{AB} $ to find $\vec{OM}$, which represents the coordinates of point M.

$$ \vec{OM} = \begin{pmatrix} 2 \ -5 \end{pmatrix} + \begin{pmatrix} -1 \ 2 \end{pmatrix} = \begin{pmatrix} 1 \ -3 \end{pmatrix} $$

So, the coordinates of point M are $ M(1, -3) $.