Let $\ell$ be the line parametrized as $(t, 2t+1, 3t+2)$ and let $P$ be the plane with equation $x+y+z = 1$.

Question

Let $\ell$ be the line parametrized as $(t, 2t+1, 3t+2)$ and let $P$ be the plane with equation $x+y+z = 1$.

This question has been asked but the answers there don't help me and I am still unsure of what to do. Please help!

What to do for (a): multiply the matrix $A$ by $\pmatrix{t\2t+1\3t+2}$ and show that the resulting point is in the plane (which is characterized by sum of components equal $1$) — J. W. Tanner, May 08 '20 at 17:07
So I did this and got $\begin{pmatrix}2\ 5t+2 \ -5t-3 \end{pmatrix}$. Then I took each of these values as x, y, and z and added them, and that added to 1. Is that enough to prove part a? — Cheez, May 08 '20 at 18:00
Yes, you took any point on the line and showed that the point $A$ maps it to is on the plane — J. W. Tanner, May 08 '20 at 18:07
much of the question (including the matrices $A$ and $B$) has been edited out, for reasons unclear to me — J. W. Tanner, May 10 '20 at 01:48

score 3 · Accepted Answer · answered May 08 '20 at 18:06

What to do for (a): multiply the matrix $A$ by $\pmatrix{t\\2t+1\\3t+2}$

and show that the resulting point is in the plane

(which is characterized by sum of components equal $1$).

What to do for (b): multiply the matrix $B$ by $\pmatrix{ x\\y\\1-x-y}$

and show that the resulting point $\pmatrix{X\\Y\\Z}$is on the line

(which is characterized by $X=\dfrac{Y-1}2=\dfrac{Z-2}3$).

1 Answers1