Orthonormal Base in $\mathbb{R}^2$

Question

i have an orthonormal base $\vec{v}_1, \vec{v}_2 \in \mathbb{R}^2$.

My book says that i can write a generic vector $\vec{v}$ (with unit length) as $$ \vec{v} = \vec{v}_1\cos(A) + \vec{v}_2 \sin(A), $$ where $A$ is the angle between $\vec{v}_2$ and $\vec{v}$.

But I have not understood why i can write $\vec{v}$ in this way using $\sin$ and $\cos$.

Answer 1

Suppose $v_1$ and $v_2$ are orthogonal basis for $\mathbb R^2$, then you could write any $v\in \mathbb R^2$ as $$v = proj_{v_1}(v) + proj_{v_2}(v)$$ where $proj_{x}(v)$ is a projection of $v$ on $x$. \begin{align*} v &= \frac{v\cdot v_1}{|v_1|}v_1+\frac{v\cdot v_2}{|v_2|}v_2\\ &= v_1|v|\cos (A) + |v|\cos\left(\frac\pi2 \pm A\right)v_2\\ &= v_1|v|\cos (A)v_1 + |v|\cos\left(\frac\pi2 \pm A\right)v_2 \end{align*} Now since $|v|=1$ and $\cos\left(\frac\pi2 \pm A\right)=\sin(A)$, the expression simplifies to $$v = v_1 \cos(A) + v_2 \sin(A)$$