Question

Verify that the space curves intersect at the given values of the parameters. Find the angle between the tangent vectors to the curves at the point of intersection. $$ \begin{array}{l} \mathbf{r}(t)=\left\langle t-2, t^{2}, \frac{1}{2} t\right\rangle, \quad t=4 \\\ \mathbf{u}(s)=\left\langle\frac{1}{4} s, 2 s, \sqrt[3]{s}\right\rangle, \quad s=8 \end{array} $$

Short answer

The space curves intersect at the point \( (2, 16, 2) \) with an angle of \( θ \) between the tangent vectors at this point.

Step-by-step solution

Verify Intersection

Evaluate the functions \(\mathbf{r}(t)\) and \(\mathbf{u}(s)\) at \(t = 4\) and \(s = 8\) respectively. If they yield the same point, the curves intersect.

Compute Tangent Vectors

Calculate the derivative of the functions \( \mathbf{r}'(t) \) and \( \mathbf{u}'(s) \) at their respective parameters. These derivatives give the tangent vectors at the point of intersection.

Calculate the Angle

The angle between the tangent vectors can be found using the dot product. The angle \( θ \) is determined by the formula \( \cos(θ) = \frac{\mathbf{r}'(t) \cdot \mathbf{u}'(s)}{||\mathbf{r}'(t)|| ||\mathbf{u}'(s)||} \)