Video: Evaluating Vector-Valued Functions

For the given function π‘Ÿ(𝑑) = 2 csc (2𝑑) 𝑖 + 3 tan (𝑑) 𝑗, evaluate π‘Ÿ(πœ‹/4).

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Video Transcript

For the given function π‘Ÿ of 𝑑 equals two csc two 𝑑 𝑖 plus three tan 𝑑 𝑗, evaluate π‘Ÿ of πœ‹ by four.

Here, π‘Ÿ is a vector-valued function. It’s a function whose range is a set of vectors. And we can evaluate π‘Ÿ of πœ‹ by four in much the usual way by substituting 𝑑 equals πœ‹ by four into each component function. For the horizontal component, the component of 𝑖, we get two csc two times πœ‹ by four. Of course, we know that csc π‘₯ is equal to one over sin π‘₯ and two times πœ‹ by four is equal to πœ‹ over two. So this becomes two over sin of πœ‹ by two. And since sin πœ‹ by two is simply one, we end up with two divided by one, which is two. We repeat this for the vertical component for 𝑗. That’s three tan of πœ‹ by four. And of course, tan of πœ‹ by four is simply one. So that’s three times one, which is three. That means that π‘Ÿ of πœ‹ by four is two 𝑖 plus three 𝑗. Contextually, this gives us the position of a point in the plane when 𝑑 is equal to πœ‹ by four.

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