If $A$ and $B$ are $(-2,-2)$ and $(2,-4)$, respectively, find the coordinates of $P$ such that $AP =\frac{3}{7} AB$ and P lies on the line segment AB.
Answer (2)
The coordinates of point P on the line segment AB, such that A P = 7 3 A B , are ( − 7 2 , − 7 20 ) .
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Calculate the vector AB by subtracting the coordinates of A from B: A B = B − A = ( 4 , − 2 ) .
Determine the vector AP by multiplying AB by the given fraction: A P = 7 3 A B = ( 7 12 , − 7 6 ) .
Find the coordinates of point P by adding the vector AP to the coordinates of A: P = A + A P = ( − 7 2 , − 7 20 ) .
State the coordinates of P: ( − 7 2 , − 7 20 ) .
Explanation
Problem Analysis We are given two points, A ( − 2 , − 2 ) and B ( 2 , − 4 ) , and we want to find the coordinates of point P on the line segment A B such that A P = 7 3 A B . This means that the vector from A to P is 7 3 of the vector from A to B .
Find vector AB First, we need to find the vector A B . To do this, we subtract the coordinates of point A from the coordinates of point B :
A B = B − A = ( 2 , − 4 ) − ( − 2 , − 2 ) = ( 2 − ( − 2 ) , − 4 − ( − 2 )) = ( 4 , − 2 ) So, A B = ( 4 , − 2 ) .
Find vector AP Next, we find the vector A P , which is 7 3 of A B :
A P = 7 3 A B = 7 3 ( 4 , − 2 ) = ( 7 3 × 4 , 7 3 × − 2 ) = ( 7 12 , − 7 6 ) So, A P = ( 7 12 , − 7 6 ) .
Find coordinates of P Now, we find the coordinates of point P by adding the vector A P to the coordinates of point A :
P = A + A P = ( − 2 , − 2 ) + ( 7 12 , − 7 6 ) = ( − 2 + 7 12 , − 2 − 7 6 ) = ( 7 − 14 + 12 , 7 − 14 − 6 ) = ( 7 − 2 , 7 − 20 ) So, P = ( − 7 2 , − 7 20 ) .
Final Answer Therefore, the coordinates of point P are ( − 7 2 , − 7 20 ) .
Examples
In architecture, when designing a bridge or a building, engineers often need to determine specific points along a structural beam or support. If they know the coordinates of the endpoints of the beam (points A and B) and need to place a support at a certain fraction of the distance between these points, they can use the same vector approach to find the exact coordinates for the support's placement (point P). This ensures the structure is properly supported and balanced.
