Inner product is a function which gives a scalar to a pair of vectors.

Inner Product − $f:mathbb{R}^n times mathbb{R}^nrightarrow kappa$ where $kappa$ is a scalar.

The basic characteristics of inner product are as follows −

Let $X in mathbb{R}^n$

• $left langle x,x right ranglegeq 0, forall x in X$

• $left langle x,x right rangle=0Leftrightarrow x=0, forall x in X$

• $left langle alpha x,y right rangle=alpha left langle x,y right rangle,forall alpha in kappa : and: forall x,y in X$

• $left langle x+y,z right rangle =left langle x,z right rangle +left langle y,z right rangle, forall x,y,z in X$

• $left langle overline{y,x} right rangle=left ( x,y right ), forall x, y in X$

Note −

• Relationship between norm and inner product: $left | x right |=sqrt{left ( x,x right )}$

• $forall x,y in mathbb{R}^n,left langle x,y right rangle=x_1y_1+x_2y_2+…+x_ny_n$

Examples

1. find the inner product of $x=left ( 1,2,1 right ): and : y=left ( 3,-1,3 right )$

Solution

$left langle x,y right rangle =x_1y_1+x_2y_2+x_3y_3$

$left langle x,y right rangle=left ( 1times3 right )+left ( 2times-1 right )+left ( 1times3 right )$

$left langle x,y right rangle=3+left ( -2 right )+3$

$left langle x,y right rangle=4$

2. If $x=left ( 4,9,1 right ),y=left ( -3,5,1 right )$ and $z=left ( 2,4,1 right )$, find $left ( x+y,z right )$

Solution

As we know, $left langle x+y,z right rangle=left langle x,z right rangle+left langle y,z right rangle$

$left langle x+y,z right rangle=left ( x_1z_1+x_2z_2+x_3z_3 right )+left ( y_1z_1+y_2z_2+y_3z_3 right )$

$left langle x+y,z right rangle=left { left ( 4times 2 right )+left ( 9times 4 right )+left ( 1times1 right ) right }+$

$left { left ( -3times2 right )+left ( 5times4 right )+left ( 1times 1right ) right }$

$left langle x+y,z right rangle=left ( 8+36+1 right )+left ( -6+20+1 right )$

$left langle x+y,z right rangle=45+15$

$left langle x+y,z right rangle=60$