How do you find parametric equations for the tangent line to the curve with the...
How do you find parametric equations for the tangent line to the curve with the given parametric equations at the point .
Answer & Explanation
We have parametric equations as functions of t for x, y and z so we can form the derivatives wrt t for each variable:
In order to find the normal at any particular point in vector space we use the Del, or gradient operator:
So for the particular point we need to establish the corresponding value of t, which by inspection is .
So for this particular point, the normal vector to the surface is given by:
So the tangent line is a line with direction that passes through the point , which therefore has the vector equation:
Which we can therefore parametrise (using ) as follows