# dot product vs inner product? [duplicate]

For $V_n$ where $x=(x_1, x_2, \ldots, x_n)$ and $y=(y_1,y_2,\ldots,y_n)$, the dot product is defined by $x_1y_1+x_2y_2+ \cdots+x_ny_n$.

In Apostol's calculs vol 2

It says that if $x=(x_1,x_2)$ and $y=(y_1,y_2)$ are any two vectors in $V_2$, define $(x \cdot y)$ by the formula

$(x \cdot y)= 2x_1y_1+x_1y_2+x_2y_1+x_2y_2$

Why is that?

Is there difference between dot and inner product?

I thought that following the dot product, $(x \cdot y)$ should be $x_1y_1+x_2y_2$.

## marked as duplicate by Jeel Shah, Jonas Meyer, user147263, Peter Woolfitt, Daniel W. FarlowApr 28 '15 at 19:20

• The dot product is another name for an inner product on $\mathbb{R}^n$ and typically refers to the one you mentioned first. Inner products are more general but share many of the same properties with the usual dot product. – Cameron Williams Sep 17 '14 at 4:50