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I'm currently working on the linear programming problem involving plotting level sets.

The first part of the problem asks me to plot the level sets of $z(x, y) = x^2 − y^2$

This wasn't too difficult for me, and the next part asks me to implicitly differentiate $x^2-y^2 = k$ and let $(x_0, y_0)$ be a point on this.

For this part I got $2yy' = 2x = dy/dx = x/y$

Part c) is where I'm a little lost. It says:

"Find an expression for a vector parallel to the tangent line at $(x_0, y_0)$ [Hint: you can use the slope you just found]"

Finding a parallel line wouldn't be to difficult as I'd just need to add a constant and keep the same slope, but I'm not exactly sure how I'd go about finding a parallel vector. As far as I know, the cross product is used to find a parallel line, but that would require 3 dimensions, right?

Any help would be greatly appreciated!

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    I think taking about parallel vector to $x^2-y^2=k$. Then you only have to worry about the $x$ and $y$ direction. – 2017-02-02
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    I haven't done much with vectors, it's been years since I took a calc class. Could you give me a quick run down of how to do that? – 2017-02-02
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    The slope is $x/y$. That means for every $\color{red}{y}$ units we go run in the $x$ direction, we go $\color{red}{x}$ units in the $y$ direction. That gives a parallel vector of $\langle \color{red}{y},\color{red}{x} \rangle$. – 2017-02-02

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