# Calc Problem: Find Angle of Intersection Between r1(t) & r2(t) at Origin

• Juntao
In summary, the problem involves finding the angle of intersection between the curves r1(t)=<t,t^2,t^3> and r2(t)=<sin t, sin 2t, t> at the origin. The hint suggests to find the tangent lines of the curves, but instead, taking the derivative of the curves and using the dot product formula, the angle is calculated to be approximately 66 degrees.
Juntao
The curves r1(t)=<t,t^2,t^3> and r2(t)=<sin t, sin 2t, t> intersect at the origin. Find their angle (acute) of interesection correct to the nearest degree. (Think! What angle are you trying to locate? Now dn't go off on a tangent.)

So that's the problem.
All I got so far is
r1(t)=t(i)+t^2(j)+t^3(k)
r2(t)=sint(i)+sin 2t(j)+t(k)

Now I'm stuck. I'm not sure where to go now.

They intersect at the origin at t=0. Now, I only know how to find the angle between two straight lines, so I would disregard the hint and take the derivative. You get r1'(t)=<1,2t,3t^2>=<1,0,0> and r2'(t)=<cost,2cos2t,1>=<1,2,1>.
Now how do you find the angle f between two vectors?
A*B=ABcosf
1=(1)sqrt(6)cosf
f=arccos(sqrt(6)/6)=66 degrees

How is that "disregarding" the hint? The purpose of the hint was to direct you to the tangent lines of the curves.

I interpreted, "Don't go off on a tangent" to mean "Tangents aren't the way to solve the problem."

## 1. How do you find the angle of intersection between two curves?

To find the angle of intersection between two curves, you first need to set the equations of the curves equal to each other. Then, solve for the variable t to find the points of intersection. Finally, plug these points into the derivative of the curves to find the slopes at the points of intersection. The angle of intersection can be calculated using the formula tan θ = (m1 - m2) / (1 + m1m2), where m1 and m2 are the slopes of the curves at the points of intersection.

## 2. What are the steps for solving a calc problem involving finding the angle of intersection between two curves?

The steps for solving a calc problem involving finding the angle of intersection between two curves are as follows:

1. Set the equations of the curves equal to each other.
2. Solve for the variable t to find the points of intersection.
3. Plug the points of intersection into the derivative of the curves to find the slopes at these points.
4. Use the formula tan θ = (m1 - m2) / (1 + m1m2) to calculate the angle of intersection, where m1 and m2 are the slopes of the curves at the points of intersection.

## 3. Can the angle of intersection between two curves be negative?

No, the angle of intersection between two curves cannot be negative. The angle is always measured as the acute angle between the two curves, so it will always be a positive value.

## 4. What is the significance of finding the angle of intersection between two curves?

Finding the angle of intersection between two curves is useful for analyzing the relationship between the curves and understanding how they intersect. It can also be used to find the angle between two lines or to determine the direction of motion at the points of intersection.

## 5. Is there a specific method or formula for finding the angle of intersection between two polar curves?

Yes, there is a specific formula for finding the angle of intersection between two polar curves. It is given by the formula tan θ = (r2sin(ϕ2) - r1sin(ϕ1)) / (r2cos(ϕ2) - r1cos(ϕ1)), where r1 and r2 are the radii of the curves and ϕ1 and ϕ2 are the angles of the curves at the points of intersection.

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