parametric eqn for tangent to (x, y, z) = (t^5, t^4, t^3)

[mathstresser]

Find the parametric equation for the tangent line to the curve

x = t^5
y = t^4
z = t^3

at the point (1, 1, 1).

I know that

r(t) = <t^5, t^4, t^3> and
r'(t) = <5t^4, 4t^3, 3t^2> and
r'(1) = <5, 4, 3>

But, I don't know how to write the parametric equation for the tangent line because we don't have t to a power.

 

[Guest]

Re: parametric eqn for tangent to (x, y, z) = (t^5, t^4, t^3

Find the parametric equation for the tangent line to the curve

x = t^5
y = t^4
z = t^3

at the point (1, 1, 1).

I know that

r(t) = <t^5, t^4, t^3> and
r'(t) = <5t^4, 4t^3, 3t^2> and
r'(1) = <5, 4, 3>

But, I don't know how to write the parametric equation for the tangent line because we don't have t to a power.

r'(t) is vector parallel to the tangent, so the tangent to the curve at r(t) an
be written:

tangent(lambda,t)=r(t)+lambda * r'(t),

lambda in R, so in this case the tangent is (1,1,1)+lambda * (5,4,3).

RonL

 

[soroban]

Re: parametric eqn for tangent to (x, y, z) = (t^5, t^4, t^3

Hello, mathstresser!

You've found all the information you need . . .

Find the parametric equation for the tangent line

to the curve: \(\displaystyle \:\begin{array}{ccc}x \,=\,t^5 \\ y\,=\,t^4 \\ z\,=\,t^3\end{array}\:\) at the point \(\displaystyle (1,\, 1,\, 1).\)

I know that: \(\displaystyle \:r(t) \:= \:\langle t^5,\,t^4,\,t^3\rangle\)

and: \(\displaystyle \,r'(t) \:= \:\langle 5t^4,\,4t^3,\,3t^2\rangle\)

and: \(\displaystyle \,r'(1) \:= \:\langle 5,\,4,\,3\rangle\;\;\) . . . Correct!

But, I don't know how to write the parametric equation for the tangent line
because we don't have t to a power. ??

The equations of the line through \(\displaystyle (x_1,\,y_1,\,z_1)\) with direction vector \(\displaystyle \vec{v}\,=\,\langle a,\,b,\,c\rangle\)

. . are: \(\displaystyle \:\begin{array}{ccc}x\:=\:x_1\,+\,ap \\ y\:=\:y_1\,+\,bp \\ z\:=\:z_1\,+\,cp\end{array}\:\) for some parameter \(\displaystyle p.\)

Go for it!