Question

Prove that the dot product of the continuous vector valued functions and is a continuous scalar function.

Short answer

Ans: It is proved that $r_1\left(t\right).r_2\left(t\right)$is a continuous scalar function.

Step-by-step solution

Step 1. Given information.

given,

and

Step 2. Solution

Consider two continuous vector functions with three components. Let

The objective is to prove that the dot product of the functions role="math" localid="1649669020226" $r_1\left(t\right)and{r}_2\left(t\right)$is a continuous scalar function.

Consider

role="math"

Step 3. Therefore,

$r_1\left(t\right)\cdot r_2\left(t\right)=x_1\left(t\right)x_2\left(t\right)+y_1\left(t\right)y_2\left(t\right)$is a scalar function.

Since $r_1\left(t\right)and{r}_2\left(t\right)$are continuous, their components $x_1\left(t\right),y_1\left(t\right),x_2\left(t\right)\text{and}{y}_2\left(t\right)$are continuous. The product and sum of the continuous functions are also continuous.

The above theory $x_1\left(t\right)x_2\left(t\right)\text{and}{y}_1\left(t\right)y_2\left(t\right)$are continuous. Hence

$x_1\left(t\right)x_2\left(t\right)+y_1\left(t\right)y_2\left(t\right)$is also continuous.

That is role="math" localid="1649669297290" $r_1\left(t\right).r_2\left(t\right)$is continuous.

Thus, $r_1\left(t\right).r_2\left(t\right)$is a continuous scalar function.