Equation of tangent line to the curve

Win_odd Dhamnekar

Win_odd Dhamnekar

Junior Member
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Aug 14, 2018
Messages
207
Hi,
How to find the equation of tangent line to the curve \(x=a\cos{\theta}, y=a\sin{\theta},z=a\theta \tan{\alpha}\) at \(\frac{\pi}{4}\)?

Solution:- Given curve is \(\vec{r}= a\cos{\theta}\hat{i}+a\sin{\theta}\hat{j}+a\theta\tan{\alpha}\hat{k}\),

Differentiating w.r.t \(\theta, \frac{d\vec{r}}{d\theta}=-a\sin{\theta}\hat{i}+a\cos{\theta}\hat{j}+a\tan{\alpha}\hat{k}\)

So the equation of a tangent line to the curve \(\vec{r}\) is \(\frac{x-\frac{a}{\sqrt{2}}}{-\frac{a}{\sqrt{2}}}= \frac{y-\frac{a}{\sqrt{2}}}{\frac{a}{\sqrt{2}}}=\frac{z-a\frac{\pi}{4}\tan{\alpha}}{a\tan{\alpha}}\) at \(\theta =\frac{\pi}{4}\)

But answer provided to me is different. How is that? I want to know where i am wrong?
If any member knows the answer to this question may reply with correct answer.
 
Dr.Peterson said:
I can't say why the provided answer is different without seeing what it is. Maybe it is equivalent to yours.
Click to expand...
Hi,
The answer provided to me is \(x-\frac{a}{\sqrt{2}}=y-\frac{a}{\sqrt{2}}=\frac{\left(z-a\frac{\pi}{4}\tan{\alpha}\right)}{\sqrt{2}\tan{\alpha}}\) at \(\theta=\frac{\pi}{4}\)
 
That answer is equivalent to yours apart from a sign. (Multiply all sides of yours by [MATH]\frac{a}{\sqrt{2}}[/MATH].)

Which of the two do you think is right?
 
Dr.Peterson said:
That answer is equivalent to yours apart from a sign. (Multiply all sides of yours by [MATH]\frac{a}{\sqrt{2}}[/MATH].)

Which of the two do you think is right?
Click to expand...
Hi,
Answer provided to me have positive x-coordinate in the equation of tangent line because \(\sin{\theta}\) can not be negative for \(\theta=\frac{\pi}{4}\). I didn't notice it, that'why i put negative sign for x-coordinate of tangent line equation.

I think this justification is sufficient.
 
I say your answer is right and theirs, as you quoted it, is wrong. If you graph the curve (a helix), you can see this.
 
Dr.Peterson said:
I say your answer is right and theirs, as you quoted it, is wrong. If you graph the curve (a helix), you can see this.
Click to expand...
Hi,
How can i plot this question and its solution in my graphing calculator 3D? I tried to plot the curve and its tangent line at \(\theta=\frac{\pi}{4}\) using following graphic controls. \(a=1, 0\leq \theta \leq 2*\pi \) or \(\frac{\pi}{2}, 0\leq \alpha\leq 2*\pi\) or \(\frac{\pi}{2}\). But i could not identify which is the curve and which is the tangent line.
Would you tell me how to plot, view and interpret this curve and its tangent line graphically?:)
 
I just plotted it in my head! I don't have a graphing calculator to try it on. There are online tools you could try, that can do it in color.
 
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