========================================================== Scalar value is notated by not-bold lower case. Multiplication of vector by scalar value: $$$y=ax=\begin{bmatrix} ax_{1}\\ax_{2}\\...\\ax_{N} \end{bmatrix}$$$ ========================================================= Inner product when 2 vectors have 0 angle : $$$c=\langle x,y \rangle=x\cdot y=x^{T}y=\sum\limits_{t=1}^{N}x_{t}y_{t}$$$ Result value of inner product is one scalar value in real number. ========================================================= When result of inner product is 0, you can say 2 vectors are perpendicular. ========================================================= Inner product when 2 vectors have angle: $$$c=||x||||y||\cos{\theta}$$$ $$$0<\theta<\pi$$$