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AL

# (a) Find the unit vectors that are parallel to the tangent line to the curve $y = 2 \sin x$ at the point $(\frac{\pi}{6}, 1)$.(b) Find the unit vectors that are perpendicular to the tangent line.(c) Sketch the curve $y = 2 \sin x$ and the vectors in parts (a) and (b), all starting at $(\frac{\pi}{6}, 1)$.

## a) $$\overline{u}_{ \|}=\frac{1}{2} i+\frac{\sqrt{3}}{2} j, \text { or }-\frac{1}{2} i-\frac{\sqrt{3}}{2} j$$b) $$\left\langle-\frac{\sqrt{3}}{2}, \frac{1}{2}\right\rangle \text { or }\left\langle\frac{\sqrt{3}}{2},-\frac{1}{2}\right\rangle$$c) The blue curve is the graph of $y=2 \sin x .$ The purple vectors are the unit tangent vectors: $\langle 1 / 2, \sqrt{3} / 2\rangle$ and $\langle- 1 / 2,-\sqrt{3} / 2\rangle$ at the point $(\pi / 6,1) .$ (The light purple line is the tangle line at $(\pi / 6,1) . )$ The red vectors are the unit vectors that are perpendicular to the tangent line: $(-\sqrt{3} / 2,1 / 2)$ and $\langle\sqrt{3} / 2,-1 / 2\rangle$at $(\pi / 6,1) .$ (The light red line is the normal line at $(\pi / 6,1) . )$

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AG

Alex G.

September 22, 2020

Can someone explain what the unit vectors is?

NH

September 22, 2020

Hey Alex, In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in.

HC

Howie C.

September 22, 2020

What is a tangent line?

ST

Samantha T.

September 22, 2020

I know this one! In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.

CR

Cam R.

September 22, 2020

Anyone else confused by the perpendicular, can someone explain?

LP

Lindsey P.

September 22, 2020

I see how that could be confusing. In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects. A line is said to be perpendicular t

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