To find vectors if we are given a function and a point

Question

I have to solve the following exercise :

Find the tangent and the vertical unit vectors in the curve at the given point (four unit vectors are requested). Also design the vectors and the curves in common shape.

$1.$ $f(x)= x^2 ,$ at $(2,4)$ and

$2.$ $x^2+y^2 =6,$ at $(2,1)$

I don't know what I have to do to find the vectors, should they be in the form $\vec{u} =a\vec{i}+b\vec{j}$ ?I tried to find the tangent lines at the given points , will they help me to find the vectors?.I think the unit vectors should be in the form of $\vec{v} =\frac{\vec{v}}{|\vec{v}|}$

Answer 1

HINT

For the first we have

• $f'(x)=2x \implies f'(2)=4 \implies T=(1,4) \quad N=(4,-1)$

For the second we have

• $x^2+y^2=6 \implies 2xdx+2ydy=0 \quad \frac{dy}{dx}=-\frac x y \implies T=(-2,1) \quad N=(1,2)$

Then find the unitary vectors $T=\frac{T}{|T|}$ and $N=\frac{N}{|N|}$.