Plane equation (weird form 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
 A: 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
$$
\{ \alpha(2,-1,-1) + \beta(1,1,0) - (-1,1,2)\}\quad \alpha,\beta \in \mathbb{R}
$$
A: 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 $t_0=s_0=0$
$$P_1(t,s)=(1,-1,-2)+t(2,-1,-1)+s(1,1,0)$$
