Can someone please help me with this problem involving vectors and points...?

Question

Let P be the point (2,3) and Q be the point (7, -4). Use vectors to find the point on the line segment joining P and Q that is 3/4 of the way from Q to P.

Answer 1

in solving that problem , you must have to use it in graphing paper .let p be the point [2,3] and Q be the point [7,-4]. so the the 2 and 7 called as x .p2 be the first x and Q7 be the second x and the p3 be the first y and Q-4 be the second y. that is what we call linear functions.  

Answer 2

Suppose that the point is A

we know that QA=3/4 QP

So, the x part of QA= 3/4 x part of QP : xA-7 = 3/4 * -5=-15/4

so xA=7-15/4=13/4

We also do that for the y part: yA-(-4)=3/4*7=21/4

so yA=21/4 - 4= 5/4

So point A is ((13/4),(5/4))

Answer 3

If you have points P and Q, then the vector P - Q points from Q to P.

P - Q = (2, 3) - (7, -4) = (2 - 7, 3 - -4) = (-5, 7)

Now, to only go 3/4 of the way, all you have to do is scale (multiply) by 3/4.

3/4 * (-5, 7) = (-15/4, 21/4) = (-3.75, 5.25)

We're not quite done, though.  Our vector (-3.75, 5.25) tells us where to go from Q.  So, in order to get to the point, we have to start at Q and then go from there.  We do this by adding our vector and Q together.

Q + (-3.75, 5.25) = (7, -4) + (-3.75, 5.25) = (7 + -3.75, -4 + 5.25) = (3.25, 1.25)

So, the point on the line segment joining P and Q that is 3/4 of the way from Q to P is (3.25, 1.25).