Question

Does anyone have a sketch for q3 of pset2: part 2?

I am struggling to visualise the problem

Answer 1

Here's the problem set:

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-b-matrices-and-systems-of-equations/problem-set-2/MIT18_02SC_pset2.pdf

Answer 2

Here is a graph using geogebra (freeware from the web)

Answer 3

The gray plane is the x-y plane. the blue plane intersects the x-y plane, forming the orange line. If you start at the origin and travel along the orange line you are walking in the direction "across the hill", i.e. not up nor down. The steepest direction would be to turn 90 degrees and head directly up (or down) the blue plane. The purple vector in the blue plane shows the steepest direction heading "up".
There may be other ways, but you can do
\[ (\vec{n} \times \hat{k}) \times \vec{n}\]

Answer 4

Thanks again for the effort and detail in the response!
Just to be clear, the orange line is the vector produced by the cross product between n and k vectors?

Answer 5

strictly speaking, n x k would point in the direction of the orange line, but out of the page, and k x n would point along the orange line in the direction into the page.

but yes, moving along the orange line is equivalent to plotting c( n x k) where c ranges from -inf to + inf