Calculate the length of the given curve

In summary, the conversation discusses the calculation of the magnitude of a vector and the formula for the length of a curve. The formula for the magnitude involves removing the basis i, j, and k, while the formula for the length of a curve includes them. The conversation also touches on the dot product of the basis vectors, with the understanding that \vec i \cdot \vec i = 1 and \vec i \cdot \vec j = 0.
  • #1
bugatti79
794
1

Homework Statement



[itex]2sin(t) \vec i +5t \vec j -2cos(t) \vec k for -5 <= t<=5[/itex]


The Attempt at a Solution



[itex]let \vec r (t)= 2sin(t) \vec i +5t \vec j -2cos(t) \vec k[/itex]

Then [itex] d \vec r(t) = 2cos(t) \vec i +5 \vec j + 2sin(t) \vec k[/itex]

[itex]|| d\vec r(t) ||= \sqrt( (2cos(t) \vec i)^2+(5 \vec j)^2+ (2 sin(t) \vec k )^2)[/itex]...?
 
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  • #2
All those i's, j's, and k's are getting in your way.
Let r(t) = <2sin(t), 5t, -2cos(t)>
dr/dt = <2cos(t), 5, 2sin(t)>
How do you calculate the magnitude of a vector?
 
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  • #3
Mark44 said:
All those i's, j's, and k's are getting in your way.
Let r(t) = <2sin(t), 5t, -2cos(t)>
dr/dt = <2cos(t), 5, 2sin(t)>
How do you calculate the magnitude of a vector?

It will be the square root of the sum of the squares

[itex]||r'(t)||= \sqrt{2cos(t)^2+5^2+(2sin(t)^2)}=\sqrt{29}[/itex] where sin^2(t)+cos^2(t)=1...?
 
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  • #4
Good! :smile:

Do you have a formula for the length of a curve?
 
  • #5
The length of the curve is the integral of this value with respect to t from -5 to 5...?

[itex]= |\sqrt{29}t|^{5}_{-5}[/itex]...?

If the above is correct, how do you justify removing the basis i,j and k suddenly to proceed with the calculation?
 
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  • #6
That's it.

The formula for the length of a vector is [itex]||x \vec i + y \vec j + z \vec k|| = \sqrt{x^2 + y^2 + z^2}[/itex]

And if you really want to include i, j, and k, consider that [itex]\vec i \cdot \vec i = 1[/itex] and that [itex]\vec i \cdot \vec j= 0[/itex]
 
  • #7
bugatti79 said:

Homework Statement


[itex]|| d\vec r(t) ||= \sqrt{( (2cos(t) \vec i)^2+(5 \vec j)^2+ (2 sin(t) \vec k )^2)}[/itex]...?

So based on your last post this is corect since the dot product of i by itself, j by itself and k by itself is one...

Thanks
 
  • #8
bugatti79 said:
So based on your last post this is corect since the dot product of i by itself, j by itself and k by itself is one...

Thanks

Yes, it is correct, but it is very unusual to write it like this.
 
  • #9
Ok, but it seems strange to suddenly drop the basis. Thanks anyhow!
 

1. What is the purpose of calculating the length of a given curve?

The length of a curve is a mathematical measurement that is useful in many different fields, such as physics, engineering, and geometry. It can help determine the area under a curve, the volume of a curved object, or the distance traveled along a curved path.

2. How do you calculate the length of a curve?

The length of a curve can be calculated using calculus, specifically through integration. This involves breaking the curve into small segments, finding the length of each segment, and then adding them together to get the total length of the curve.

3. Can the length of a curve be calculated without using calculus?

Yes, there are other methods for calculating the length of a curve without using calculus. One method is to approximate the curve using straight lines and then use the Pythagorean theorem to find the length of each segment. Another method is to use a ruler or measuring tape to physically measure the curve's length.

4. What are the limitations of calculating the length of a curve?

Calculating the length of a curve can be a complex and time-consuming process. It also requires a clear understanding of calculus and mathematical concepts. Additionally, some curves may be impossible to measure accurately, such as infinitely long curves or curves with constantly changing slopes.

5. How can the length of a curve be applied in real-world situations?

The length of a curve has many practical applications in various fields. For example, it can be used in engineering to determine the amount of material needed to build a curved structure. It can also be used in physics to calculate the work done by a force moving along a curved path. In geometry, the length of a curve can help determine the area under a curve or the perimeter of a curved shape.

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