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given,

and

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"

$r_1\left(t\right)鈰�r_2\left(t\right)=x_1\left(t\right)x_2\left(t\right)+y_1\left(t\right)y_2\left(t\right)$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.