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Find a vector function that represents the curve of intersection of the two surfaces.

The hyperboloid $ z = x^2 - y^2 $ and the cylinder $ x^2 + y^2 = 1 $

$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\cos 2 t \mathbf{k}, 0 \Leftrightarrow t=2 \pi$

Vector Functions

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Boston College

For this exercise, we have to find the vector function that represents the curve of intersection of two surfaces. And those surfaces are X squared plus y squared equals one a cylinder n z squared r Z equals x squared minus y squared. So since we know that co sign squared of theta plus sine squared of theta equals one, this implies that X equals co sign of T and why equals sanity? Then we can substitute this equation in for the Z equals X squared plus y squared here. So what we end up getting is that Z equals ico sine squared of T minus the sine squared of T, which is equal to co sign of two t Then that gives us the vector function that we're looking for, which is our f t equals co sign of T I flash sign he Jay plus co sign to t. K. Because we got this Z value from combining them. And we already determined the X and the Y values, which is r i n r j. Based on the fact, um, that we knew that coastline squared of data post sine squared of theta equals one

California Baptist University

Vector Functions