Hello, Jenny20!

Given: .r(t) .= .t²i+ tj+3t³k

Find an equation for the tangent line tor(t) at (1,1,3).

The tangent vector is the derivative ofr(t):

. .r'(t) .= .2ti+j+ 9t²k

At t = 1: .r'(1) .= .2i+j+ 9k.= .< 2,1,9 >

The parametric equations of the tangent line are:

. . x .= .1 + 2p

. . y .= .1 + p . . for some parameterp

. . z .= .3 + 9p