Module 2: The Dot Product

Let LaTeX: x=(x_1,x_2,\ldots,x_n)x=(x1,x2,,xn) and LaTeX: y=(y_1,y_2,\ldots, y_n)y=(y1,y2,,yn) be two vectors in LaTeX: \mathbb{R} ^{n} Rn. Then the dot product of LaTeX: xx and LaTeX: yy is:

LaTeX: x \cdot y = x_1y_1+x_2y_2+ \cdots + x_ny_n = \sum_{k=1}^n x_ky_k.xy=x1y1+x2y2++xnyn=nk=1xkyk.

It is important to note that the dot product of two vectors is a real number (and not a vector)!

Examples:

1. Let LaTeX: x=(1,2)x=(1,2) and LaTeX: y=(-1,4)y=(1,4). Then

LaTeX:  x \cdot y = (1,2) \cdot (-1,4) = 1 \cdot -1 + 2 \cdot 4 = -1+8 = 7.xy=(1,2)(1,4)=11+24=1+8=7.

2. Let LaTeX: x=(1,2,-1)x=(1,2,1) and LaTeX: y=(-1,-2,t)y=(1,2,t). Find all values of LaTeX: tt which make LaTeX: x \cdot y=0xy=0.

LaTeX: x \cdot y = -1-4-t = 0 \Rightarrow t=-5.xy=14t=0t=5.