# Vector as a function of theta

#### sandra123

Express the vector $$\displaystyle \vec{OA}$$ and $$\displaystyle \vec{AB}$$ as a function av θ. Suppose that the angle between OA and AB is 90°.
I have begun to define the unitvector: $$\displaystyle \underline{e}_{OA}=\begin{bmatrix}-sin\theta\\ -cos\theta\\0\end{bmatrix}\leadsto \vec{OA}=40\begin{bmatrix}-sin\theta\\ -cos\theta\\0\end{bmatrix}$$. I have now a relation for the vector OA as a function of θ. My question is if the overall position of the line segment AB matter at all? I think that AB rotates similarly by an angle theta. However I'm not sure how I can get a relation with the vector AB as a function of θ.

#### studiot

hi sandra you are nearly there.

You are correct Ab rotates the same as OA, but from the horizontal, not the vertical.

See if you can carry on from that

topsquark

#### sandra123

So am I looking for the change in y-direction?

#### studiot

Keep going you will get there.

#### sandra123

To get from A to B I multiplied the unit vector with its magnitude and it is the same vector as in OA?

#### sandra123

I'm assuming that the angle theta is the same for the both triangles, how can we know that for sure?

#### studiot

I'm assuming that the angle theta is the same for the both triangles, how can we know that for sure?
The horizontal and vertical dashed lines are perpendicular, as I noted in the diagram.

I also showed the second angle as (90 - θ) - the angle at B - in the original triangle from the vertical with θ - the angle at O - in it.
So the triangle between AB and the dashed horizontal line contains a right angle and the angle at B or (90 - θ)
So its third angle - the angle at A must be θ.