Question

Find all unit vectors that are parallel to the line tangent to the curve y = 2 sin x at the point (π/6, 1). (b) Find the unit vectors that are perpendicular to that tangent line. (c) Sketch the curve y = 2 sin x and the vectors from parts (a) and (b), all starting at the point (π/6, 1).

Answer 1

Answer:

For (a) ±< cos ( π /6) , sin( π /6 ) > = ± < 1 /2 , √ 3 /2 > For (b)=   < √ 3 /2 ,- 1 /2 > For (c) see attachment.

Step-by-step explanation:

y'atx=π/6 is 2cos(π/6)=√3.

This direction θ=ψ is given by tanψ=√3.

Inversely, ψ=tan−1√3 is π/6. For the opposite direction ,

it is π+π/6.

The unit vector in the direction θ=π/6 is

<cos(π/6),sin(π/6)>.

For the opposite direction, it is

<cos(π+π/6),sin(π+π/6)>

=<−cos(π/6),−sin(π/6)> .

For Part (b)

That is, the slope of the tangent line is √3. So

(1,√3)

gives the direction of the line and

(√3,−1)

gives the direction of the perpendicular line.

(Note that (b,−a)⊥(a,b) for any vector.)

For Part C

See the link