Plane equation (weird form question)

Question

So I have this equation in a homework:

$P_1 = \{Q \in \Bbb R^3|\ (Q\ +\ (-1,1,2)) \in \langle(2,-1,-1),(1,1,0) \rangle\}$

So, my question is, how can this be a plane?, the dot product is going to be a number right?, am I missing something

Answer 1

The plane here is defined by the set $\{Q\in\mathbb{R}^3\}$ where $Q$ satisfies the given conditions.

There isn't a dot product in that description. In this case $$ \langle(2,-1,-1),(1,1,0) \rangle $$ indicates all linear combinations of the 2 vectors i.e. $$ \alpha(2,-1,-1) + \beta(1,1,0) \quad \alpha,\beta \in \mathbb{R} $$ This notation can be used with any number of vectors e.g. $$ \langle(2,-1,-1) \rangle $$ would indicate a line ( all 'linear cobinations' $\alpha(2,-1,-1)$)

Thus your plane is given by the set $$ Q+(-1,1,2) \in\{ \alpha(2,-1,-1) + \beta(1,1,0)\}\quad \alpha,\beta \in \mathbb{R} $$ or $$ \{ \alpha(2,-1,-1) + \beta(1,1,0) - (-1,1,2)\}\quad \alpha,\beta \in \mathbb{R} $$

Answer 2

The symbol

$$\langle(2,-1,-1),(1,1,0) \rangle$$

is often used to indicate also the span of the vectors thus it corresponds to the plane through the origin

$$P(t,s)=t(2,-1,-1)+s(1,1,0)$$

and the plane $P_1$ is in the form

$$P_1(t,s)=v_0+t(2,-1,-1)+s(1,1,0)$$

with

$$v_0=(1,-1,-2)+t_0(2,-1,-1)+s_0(1,1,0)$$

for some $t_0,r_0 \in \mathbb{R}$ and notably for $t_0=s_0=0$

$$P_1(t,s)=(1,-1,-2)+t(2,-1,-1)+s(1,1,0)$$