Q. 25 - Calculus

Question

Explain why the graph of every vector-valued function $r\left(t\right)=\left(\cos t, \sin t, \cos t\right)$ lies on the intersection of the two cylinders $x^2+y^2=1 \mathrm{and} y^2+z^2=1.$

Step-by-Step Solution

Answer

The graph of every vector-valued function $r\left(t\right)=\left(\cos t, \sin t, \cos t\right)$ lies on the intersection of the two cylinders $x^2+y^2=1 \mathrm{and} y^2+z^2=1$ because r(t) satisfies both cylinders.

Step 1. Given Information.

The given vector-valued function is $r\left(t\right)=\left(\cos t, \sin t, \cos t\right)$ and two cylinders are $x^2+y^2=1 \mathrm{and} y^2+z^2=1.$

Step 2. Explanation.

To explain why the graph of every vector-valued function lies pon the intersection of the given two cylinders, take the cylinder $x^2+y^2=1.$

Now, substitute $x=\cos t \mathrm{and} y=\sin t$in the above cylinder equation,

$$\begin{gathered} x^2+y^2=1 \\ {\cos }^{2}t+{\sin }^{2}t=1 \\ 1=1 \end{gathered}$$

It is true.

Now, take the cylinder $y^2+z^2=1.$

Substitute $y=\sin t \mathrm{and} z=\cos t$ in the above cylinder equation,

$$\begin{gathered} y^2+z^2=1 \\ {\sin }^{2}t+{\cos }^{2}t=1 \\ 1=1 \end{gathered}$$

It is true.

Thus, r(t) satisfies both cylinders.

Hence the graph of every vector-valued function $r\left(t\right)=\left(\cos t, \sin t, \cos t\right)$ lies on the intersection of the two cylinders $x^2+y^2=1 \mathrm{and} y^2+z^2=1.$