Q. 18 - Calculus

Question

Compute the cross product of the vector functions $r_1\left(t\right)=\left(x_1\left(t\right), y_1\left(t\right)\right) \mathrm{and} \left(r_2\left(t\right)=x_2\left(t\right), y_2\left(t\right)\right)$ by thinking of ${\mathrm{ℝ}}^2$ as the xy-plane in ${\mathrm{ℝ}}^3.$ That is, let $r_1\left(t\right)=\left(x_1\left(t\right), y_1\left(t\right), 0\right) \mathrm{and} r_2\left(t\right)=\left(x_2\left(t\right), y_2\left(t\right), 0\right),$ and take the cross product of these vector functions.

Step-by-Step Solution

Answer

The cross product of the given vector functions is $\left(x_1\left(t\right)y_2\left(t\right)-y_1\left(t\right)x_2\left(t\right)\right)k \mathrm{or} \left(0,0,x_1\left(t\right)y_2\left(t\right)-y_1\left(t\right)x_2\left(t\right)\right).$

Step 1. Given Information.

The given vector functions are $r_1\left(t\right)=\left(x_1\left(t\right), y_1\left(t\right), 0\right) \mathrm{and} r_2\left(t\right)=\left(x_2\left(t\right), y_2\left(t\right), 0\right).$

Step 2. Find the cross product.

Let's find the cross product of two vectors, 

$$\begin{gathered} =r_1\left(t\right)\times r_2\left(t\right) \\ =\left|\begin{array}{ccc}i& j& k\\ x_1\left(t\right)& y_1\left(t\right)& 0\\ x_2\left(t\right)& y_2\left(t\right)& 0\end{array}\right| \\ =i\left[0\right]-j\left[0\right]+k\left[x_1\left(t\right)y_2\left(t\right)-y_1\left(t\right)x_2\left(t\right)\right] \\ =\left(x_1\left(t\right)y_2\left(t\right)-y_1\left(t\right)x_2\left(t\right)\right)k \end{gathered}$$