Question

The curves r1(t) = 4t, t², t³ and r2(t) = sin(t), sin(3t), 5t intersect at the origin. find their angle of intersection, , correct to the nearest degree.

Answer 1

To find the angle of intersection between the curves at the origin, calculate the tangent vectors, use the dot product to find the cosine of the angle, and then use the arccos function to determine the angle θ, rounding to the nearest degree.

Step-by-step explanation:

To find the angle of intersection between the curves r1(t) = (4t, t², t³) and r2(t) = (sin(t), sin(3t), 5t) at the origin, we first determine the tangent vectors of each curve at the point of intersection. The tangent vector is found by taking the derivative of the position vector r(t) with respect to t. Differentiating both r1(t) and r2(t), we get:

• r1'(t) = (4, 2t, 3t²)
• r2'(t) = (cos(t), 3cos(3t), 5)

At the origin, t=0, so the tangent vectors become:

• r1'(0) = (4, 0, 0)
• r2'(0) = (1, 0, 5)

The angle θ between the tangent vectors can be found using the dot product formula:

cos(θ) = ​{r1'(0) · r2'(0)} /

Substituting the vectors into the formula:

cos(θ) = {(4 · 1 + 0 · 0 + 0 · 5)} / { √(4² + 0² + 0²) √(1² + 0² + 5²) }

cos(θ) = 4 / √(16 × 26)

cos(θ) = 4 / √416

θ = arccos(4 / √416)

Calculate the angle and round to the nearest degree to get the angle of intersection.