Question

Find the unit tangent vector \(\mathbf{T}(t)\) and find a set of parametric equations for the line tangent to the space curve at point \(\boldsymbol{P}\) $$ \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+t \mathbf{k}, \quad P(2,0,0) $$

Short answer

The unit tangent vector at \(2,0,0\) is \(\mathbf{T}(t) = -\frac{2 \sin t}{2} \mathbf{i} + \frac{2 \cos t}{2} \mathbf{j}+ \frac{t}{t} \mathbf{k}\) . The parametric equation for the tangent line to the given space curve at the given point is \(\mathbf{r}(t) = (2, 0, 0) + t(-\frac{2 \sin t}{2} \mathbf{i} + \frac{2 \cos t}{2} \mathbf{j}+ \frac{t}{t} \mathbf{k})\). Note the solution may vary depending on how you simplify the expressions.

Step-by-step solution

Compute the derivative of the curve

First step involves finding the derivative of the given parametric equation, which gives us the velocity vector. That could be done by differentiating each component of \(\mathbf{r}(t) = 2 \cos t \mathbf{i} + 2 \sin t \mathbf{j} + t \mathbf{k}\) with respect to \(t\).

Find the magnitude of the derivative

Calculate the magnitude or norm of the derivative vector. This is achieved by squaring each component of the derivative vector, summing these, and then taking the square root.

Compute the unit tangent vector

To find the unit tangent vector, divide the derivative vector by its magnitude. The unit tangent vector will give the direction of the curve.

Find the value of 't' at point P

Substitute the coordinates of point \(P(2,0,0)\) into the given parametric equation, and solve for \(t\). It's important because we are finding the tangent line at this specific point.

Calculate the unit tangent vector at point \(P\)

Substitute the value of \(t\) obtained in step 4 into the unit tangent vector equation found in step 3. The resulting vector is the unit tangent vector at point \(P\).

Formulate the equation of the tangent line

The tangent line at point \(P\), can be formulated by using the formula \(\mathbf{r}(t) = \mathbf{r_0} + t \mathbf{T}\), where \(\mathbf{r_0}\) is the position vector of \(P\), and \(\mathbf{T}\) is the direction or the unit tangent vector at \(P\).