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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
