Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t)=cos(t)I -cos(t)j+sin(t)k
I don't know what to do. I let x=cos(t), y=-cos(t) and z= sin(t). Should I let t be any number and get the equal values for cosine and sine and start ploting or there something else to do?
They said sketch it. Start at t = 0, do t in degrees so you can see change easily
t |x y z
0| 1 , -1 , 0
1| .99 ,-.99, .02
9| .99,-.99,.15
20|.94,-.94, .34 note z changing faster than others
now look at change of r with change in t
dr/dt = -sin t i + sin t j + cos t k
at t = 0
dr/dt = 0 i + 0 j + 1 k
so at t = 0, r is headed up totally in z
To sketch the curve with the given vector equation, you can start by parameterizing the equation. In this case, the parameter is t.
Let's consider the given vector equation:
r(t) = cos(t)i - cos(t)j + sin(t)k
Step 1: Parameterize the equation:
Let x = cos(t), y = -cos(t), and z = sin(t).
So we have:
r(t) = xi + yj + zk
Step 2: Replace x, y, and z with their parameterized values:
r(t) = cos(t)i - cos(t)j + sin(t)k
Now, let's focus on plotting the curve:
1. Start by varying the parameter t. Choose different values for t to evaluate the corresponding position vector r(t).
2. Compute the values of x, y, and z for each chosen value of t.
3. Plot these points in a 3D coordinate system using the Cartesian coordinates (x, y, z).
The direction of t increasing can be noted by observing how the curve moves in the plot. For example, if the curve moves in a clockwise direction as t increases, you can indicate the direction with an arrow pointing clockwise.
Remember, for this specific curve, the three coordinate functions are periodic. So it's better to choose values of t that span one full period, such as t = 0 to t = 2π or t = -π to t = π. This will give you a better understanding of the overall shape of the curve.
By following these steps and plotting the points, you should be able to sketch the curve.
To sketch the curve with the given vector equation, you can follow these steps:
1. Substitute the given coordinates for the curve into their corresponding axes. In this case, you have:
- x = cos(t)
- y = -cos(t)
- z = sin(t)
2. Since the equations for x, y, and z are all in terms of t, you can let t be any value you wish. However, keep in mind that the parameter t determines the position on the curve. So, choosing different values for t will give you different points on the curve.
3. To plot the curve, choose a range of values for t that cover a reasonable portion of the curve. For example, you could use the range t = 0 to t = 2π, which will correspond to one full revolution of the curve.
4. For each value of t within your chosen range, substitute it into the equations for x, y, and z to get the corresponding coordinates.
5. Plot the points you obtained from step 4 in 3D space. Connect these points together, and you will have the sketch of the curve.
6. Finally, to indicate the direction in which t increases, you can use arrows along the curve. These arrows will point in the same direction as the increasing values of t.
Note: Depending on the desired level of detail, you may need to plot more points on the curve by choosing additional values for t within your chosen range.