Question

Use a computer algebra system to graph the space curve. Then find \(\mathbf{T}(t)\) and find a set of parametric equations for the line tangent to the space curve at point \(P\). Graph the tangent line. $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+4 \sin t \mathbf{j}+\frac{1}{2} t \mathbf{k}, \quad P(0,4, \pi / 4) $$

Short answer

The tangent vector \( \mathbf{T}(t) = -3 \sin t \mathbf{i} + 4 \cos t \mathbf{j} + \frac{1}{2} \mathbf{k} \) evaluated at \( t = \pi / 4 \) is \( \mathbf{T}(\pi / 4) = -3 \sin(\pi / 4) \mathbf{i} + 4 \cos(\pi / 4) \mathbf{j} + \frac{1}{2} \mathbf{k} \). The parametric equations for the tangent line at point \( P(0,4, \pi / 4) \) are \( x = t - 3 \sqrt{2}t \), \( y = 4 + 2 \sqrt{2}t \), and \( z = \pi / 4 + t/2 \).

Step-by-step solution

Determine the derivative of \( \mathbf{r}(t) \)

Differentiate each component function of \( \mathbf{r}(t) \) with respect to \( t \) to obtain \( \mathbf{r}'(t) \) which is also the tangent vector \( \mathbf{T}(t) \). So, \( \mathbf{T}(t) = \mathbf{r}'(t) = -3 \sin t \mathbf{i} + 4 \cos t \mathbf{j} + \frac{1}{2} \mathbf{k} \)

Evaluate \( \mathbf{T}(t) \) at \( t = \pi / 4 \)

Substitute \( t = \pi / 4 \) into the expression obtained in Step 1. The resulting vector \( \mathbf{T}(\pi / 4) \) is the direction of the tangent line at P. So, \( \mathbf{T}(\pi / 4) = -3 \sin(\pi / 4) \mathbf{i} + 4 \cos(\pi / 4) \mathbf{j} + \frac{1}{2} \mathbf{k} \).

Find parametric equations for the tangent line

Use \( \mathbf{r}_0 + t \mathbf{T}(t) \), where \( \mathbf{r}_0 \) is the position vector of the point of tangency. Substitute \( \mathbf{r}_0 = (0,4,\pi / 4) \), and \( \mathbf{T}(t) = \mathbf{T}(\pi / 4) \) obtained in Step 2. The parametric equations are \( x = t - 3 \sqrt{2}t \), \( y = 4 + 2 \sqrt{2}t \), and \( z = \pi / 4 + t/2 \)

Graph the space curve and the tangent line

The space curve is captured by plotting all possible vectors \( \mathbf{r}(t) \) for \( t \) in a given range. Use the parametric equations found in step 3 to draw the line tangent to the curve at point P. Plot these on the same graph.