Anyone Good With Calculus 3?

Question

have a feeling I'm doing something wrong, but here's the question:

You start walking at a point with coordinates (2,-2). You arrive at point (4,2). If you begin walking in the direction of the vector 4i + 2j and change direction only once, when you turns at a right angle, what are the coordinates of the point where you make the turn?

I'm not sure if it's the phrasing that's confusing me, but exactly how do I got about solving this problem?

Answer 1

Your initial position is 2i - 2j. You walk some distance in the direction of (4i + 2j). Then you move at a right angle to this direction for some distance, and end up at position 4i + 2j.

A vector in the plane perpendicular to (4i + 2j) is (2i - 4j) (switch the components, and multiply one of them by -1 -- note that the dot product is zero). There are two choices; you can use a diagram to pick the correct one, or let the math do it for you.

So we have

2i - 2j + m(4i + 2j) + n(2i - 4j) = 4i + 2j

where m and n are numbers (since the direction vectors aren't unit vectors, m and n are not the distances in each direction).

This is equivalent to the two equations

2 + 4m + 2n = 4

-2 + 2m - 4n = 2

or

4m + 2n = 2

2m - 4n = 4

whose solution is m = 4/5, n = -3/5 (the negative value of n tells me that the second leg of the trip was in the direction -(2i - 4j), not (2i - 4j) as I arbitrarily chose it.)

The coordinates of the point where you make the turn are

2i - 2j + m(4i + 2j) = 2i - 2j + (4/5)(4i + 2j) = (26/5, -2/5)