Orthogonal unit vectors If $u_1$ and $u_2$ are orthogonal unit vectors and v = $au_1$ + $bu_2$ (a and b being scalars), find $v\cdot u_1$ .
I know that by the dot product properties, $u_1 \cdot u_2$ = 0. The problem is that I can't figure out a way to use this information wise. I'd love to understand this so please help me solve it. Thanks
Well, we have $$\vec{v} = a\vec{u_1} + b\vec{u_2} \implies \\\begin{align} \vec{v} \cdot \vec{u_1} &= \left(a\vec{u_1} + b\vec{u_2}\right) \cdot \vec{u_1} \\&= a\left(\vec{u_1} \cdot \vec{u_1}\right) + b\left(\vec{u_2} \cdot \vec{u_1}\right) \\&= a\,\,, \end{align}$$ where the last equality applies the fact that the vectors are orthonormal (orthogonal unit vectors). Recall that the dot product of a unit vector with itself is $1$ and since these unit vectors are orthogonal, as you said, $\vec{u_1} \cdot \vec{u_2} = \vec{u_2} \cdot \vec{u_1} = 0$.