Rewrite the vector equation 𝐫(𝑡)=(−3+𝑡)𝐢+(1+3𝑡)𝐣+(−5−𝑡)𝐤 r ( t ) = ( − 3 + t ) i + ( 1 + 3 t ) j + ( − 5 − t ) k as the corresponding parametric equations for the line.

Question

Rewrite the vector equation 𝐫(𝑡)=(−3+𝑡)𝐢+(1+3𝑡)𝐣+(−5−𝑡)𝐤 r ( t ) = ( − 3 + t ) i + ( 1 + 3 t ) j + ( − 5 − t ) k as the corresponding parametric equations for the line.

x(t)

y(t)

z(t)

Answer 1

𝐫(𝑡)=(−3+𝑡)𝐢+(1+3𝑡)𝐣+(−5−𝑡)𝐤

is the parametric form of line (parameter is t, which I assume is any Real number).

x(t)=-3+t

y(y)=1+3t

z(t)=-5-t

another representation of the same line is called symmetric

here: t=x+3, 1+3t=y ,,,,y=3x+10,,,x=y/3-10/3

z=-5-t,,,,z=-8-x,,,x=-8-z

x=y/3-10/3=8-z

same as

3x=y-10=-24-3z

this double equality is called the symmetric form of line.

check, 3x=3t - 9

y-10=3t - 9

-24-3z==3t - 9