Question

What is the dot product of the vector functions $r_1\left(t\right)=\left(x_1\left(t\right),y_1\left(t\right)\right)\mathrm{and}{r}_2\left(t\right)=\left(x_2\left(t\right),y_2\left(t\right)\right)?$

Short answer

The dot product of the given two vector functions is$x_1\left(t\right)x_2\left(t\right)+y_1\left(t\right)y_2\left(t\right).$

Step-by-step solution

Step 1. Given Information.

The given vector functions are$r_1\left(t\right)=\left(x_1\left(t\right),y_1\left(t\right)\right)\mathrm{and}{r}_2\left(t\right)=\left(x_2\left(t\right),y_2\left(t\right)\right).$

Step 2. Find the dot product.

Let's find the dot product,

$$\begin{gathered} r_1\left(t\right)·r_2\left(t\right) \\ =\left(x_1\left(t\right),y_1\left(t\right)\right)·\left(x_2\left(t\right),y_2\left(t\right)\right) \\ =\left(x_1\left(t\right)i+y_1\left(t\right)j\right)·\left(x_2\left(t\right)i+y_2\left(t\right)j\right) \\ =x_1\left(t\right)x_2\left(t\right)\left(i·i\right)+x_1\left(t\right)y_2\left(t\right)\left(i·j\right)+y_1\left(t\right)x_2\left(t\right)\left(j·i\right)+y_1\left(t\right)y_2\left(t\right)\left(j·j\right) \\ \left[\mathrm{Since}i·i=1,i·j=0,j·i=0,j·j=1\right] \\ =x_1\left(t\right)x_2\left(t\right)+y_1\left(t\right)y_2\left(t\right) \end{gathered}$$

Thus, the product of two vector functions is a scalar function.