Vectors in 2-space and 3-space?

Question

Let P be the point (1, 3, 7). If the point (4, 0, -6) is the midpoint of line segement connecting P and Q, what is Q?

I know that v= (x2-x1, y2 - y1, z2 - z1)

What now?

Answer 1

Your vector is in 3D space.

An order of imagining 3D points greatly help to solve problems like this

x...y...z

1...3...7 is start point.
4...0..-6 is mid point

3...3...13 is x, y and z differences of start and mid points.

7...3...-19 is the end point of your question. (We have got it by simple subtraction of 3...3...13 from respective mid point co-ordinates 4...0...-6)

Please note that respective co-ordinates difference and subtraction are differ rent and choosing it logically is the skill needed to solve problem like these.

Start ....................middle... .... end
(1, 3, 7)............. (4, 0, -6)..............(7,3,-19)

How to check?

(1, 3, 7).......(1)
(4, 0, -6).......(2)
------------
(3, 3, 13) .... difference of (1) and (2)

(4, 0, -6).......(2)
(7,3,-19).......(3)
------------
(3, 3, 13)....difference of (2) and (3)

coordinate differences being same it relates straight line and that (4,0,-6) is mid point!

Answer 2

Given point P(1, 3, 7). If the point M(4, 0, -6) is the midpoint of line segement connecting P and Q, what is Q?

Calculate vector PM.

PM = = <4-1, 0-3, -6-7> = <3, -3, -13>

Q = P + 2(PM) = <1, 3, 7> + 2<3, -3, -13>
Q = <7, -3, -19>