Graph the curve traced by the vector function𝒓 𝑡 = 2 cos 𝑡 𝒊 + 2 sin 𝑡 𝒋 + 3 𝒌

To graph the curve traced by the vector function 𝒓(𝑡) = 2 cos(𝑡)𝒊 + 2 sin(𝑡)𝒋 + 3𝒌, follow these steps:

1. Recognize the form of the vector function: The function is in the form of a helix, a type of curve in three-dimensional space. The i and j components represent a circle in the xy-plane, and the k component represents a constant in the z-direction.

2. Identify the radius of the circle in the xy-plane: The coefficients of the cos(𝑡) and sin(𝑡) functions are both 2, so the radius of the circle is 2.

3. Identify the height of the helix: The coefficient of the k component is 3, so the helix is constantly at a height of 3 in the z-direction.

4. Plot the curve: Using a 3D graphing tool, plot the curve for a range of t values. The x and y coordinates will trace out a circle of radius 2 in the xy-plane, and the z-coordinate will always be 3.

5. Interpret the graph: The graph represents a circle of radius 2 in the xy-plane, lifted 3 units in the z-direction. This forms a helix that circles around the z-axis at a constant height.

Remember, the graphing part might be a bit tricky without a 3D graphing tool. You can use online tools or software like GeoGebra, Desmos, or a graphing calculator that supports 3D graphing.