Find Tangent Line to Path: x(t)=4costi-3sintj+5tk, t=\pi/3

In summary, the equation for the path is x(t) = 4cos(t)i - 3sin(t)j + 5tk. The value of t for which the tangent line needs to be found is π/3. To find the tangent line to a given path, we need to find the derivative of the path with respect to t, substitute the given value of t to find the slope, and use the point-slope formula to find the equation of the tangent line. The coordinates of the point where the tangent line touches the path can be found by substituting the given value of t into the equation for x(t), which in this case is (2, -3, 5). The path and tangent line can
  • #1
rgalvan2
30
0
Find an equation for the line tangent to the given path at the indicated value for the parameter.

x(t)=4costi-3sintj+5tk, t=[tex]\pi[/tex]/3

So what I did here was take x'(t) and then plugged in [tex]\pi[/tex]/3 after that to get an equation containing i, j, and k.
x'(t)= -4sinti-3costj+5k
x'([tex]\pi[/tex]/3)=-4[tex]\sqrt{3}[/tex]/2i-3/2j+5k
Am I doing this right? Calc III is confusing me so much. Any help would be appreciated. This is due Thursday. Thanks!
 
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  • #2
Looks all right so far.
 
  • #3
Alright sweet thanks a lot!
 

1. What is the equation for the path given by x(t)?

The equation for the path is x(t) = 4cos(t)i - 3sin(t)j + 5tk.

2. What is the value of t for which the tangent line needs to be found?

The value of t is π/3.

3. How do you find the tangent line to a given path?

To find the tangent line to a given path, we need to first find the derivative of the path with respect to t. Then, we can substitute the given value of t into the derivative to find the slope of the tangent line. Finally, we can use the point-slope formula to find the equation of the tangent line.

4. What are the coordinates of the point where the tangent line touches the path?

The coordinates of the point where the tangent line touches the path can be found by substituting the given value of t into the equation for x(t). In this case, the coordinates are (2, -3, 5).

5. Can you graph the path and tangent line on a 3D coordinate system?

Yes, the path and tangent line can be graphed on a 3D coordinate system. The path will take the shape of a helix, while the tangent line will be a straight line touching the helix at the given point.

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