Question

The curves r1(t) = 5t, t², t⁴ and r2(t) = sin(t), sin(2t), 5t intersect at the origin. Find their angle of intersection, correct to the nearest degree.

Answer 1

To find the angle of intersection between curves r1(t) and r2(t) at the origin, calculate the tangent vectors by differentiating, then use the dot product formula and inverse cosine function to find the angle in radians and convert it to degrees.

Step-by-step explanation:

To find the angle of intersection between the curves r1(t) and r2(t), we need to calculate the angle between their tangent vectors at the point of intersection. The tangent vectors can be found by differentiating each component of r1(t) and r2(t) with respect to t.

For r1(t) = 5t, t², t⁴, the tangent vector at the origin (t = 0) is given by the derivative:
r1'(t) = (5, 2t, 4t³) and at the origin r1'(0) = (5, 0, 0).

Similarly, for r2(t) = sin(t), sin(2t), 5t, the tangent vector at the origin is given by the derivative:
r2'(t) = (cos(t), 2cos(2t), 5) and at the origin r2'(0) = (1, 2, 5).

To find the angle θ between the two vectors, we use the dot product formula:
cos(θ) = (r1'(0) ⋅ r2'(0)) / (|r1'(0)| |r2'(0)|).

After calculating the dot product and magnitude of both vectors, we use the inverse cosine function to find θ, and finally convert the result to degrees to find the angle of intersection correct to the nearest degree.