How to Calculate Unit Vector for Mountain Contourlines?

[hexa]

I'll be greatful for any hint.

Imagine you walk over the contourlines of the map of a mountain (really! that's the question) with a gradient of h(x,y)=2xy, x^2). You are at point (1,3) and you want to walk downslope at an angle of 45 degrees. calculate the unit vector in order to find out in which direction to walk.

Hexa

 

[HallsofIvy]

What's this doing in precalculus? This is a partial derivative problem. If the gradient is (2xy, x²), then the derivative in the direction of unit vector (a, b) is just the dot product, 2xya+ x²b. If you are walking downslope at an angle of 45 degrees, then the slope must be tan(45)= 1. You want 2xya+ x²b= 1 and, of course, a²+ b²= 1. Solve for a and b in terms of x and y.

 

[hexa]

Thanks a lot for your help. I'll work on with this.

Calculus: not a term used in this country, so I'm not quiet sure where the border is between precalculus and calculus.

 

[HallsofIvy]

Then you might say "analysis" or "applied analysis". Essentially, derivatives and integrals are calculus. The basics of limits might be in calculus or pre-calculus.