Vectors in Space

Find the canonical name for each vector.

• the vector from $(2,1)$ to $(4,2)$ in $\mathbb{R}^{2}$

• the vector from $(3,3)$ to $(2,5)$ in $\mathbb{R}^{2}$

• the vector from $(1,0,6)$ to $(5,0,3)$ in $\mathbb{R}^{3}$

• the vector from $(6,8,8)$ to $(6,8,8)$ in $\mathbb{R}^{3}$

Decide if the two vectors are equal.

• the vector from $(5,3)$ to $(6,2)$ and the vector from $(1,-2)$ to $(1,1)$

• the vector from $(2,1,1)$ to $(3,0,4)$ and the vector from $(5,1,4)$ to $(6,0,7)$

Does $(1,0,2,1)$ lie on the line through $(-2,1,1,0)$ and $(5,10,-1,4)$?

• Describe the plane through $(1,1,5,-1)$, $(2,2,2,0)$, and $(3,1,0,4)$.

• Is the origin in that plane?

Give a vector description of each.

• the plane subset of $\mathbb{R}^{3}$ with equation $x-2y+z=4$

• the plane in $\mathbb{R}^{3}$ with equation $2x+y+4z=-1$

• the hyperplane subset of $\mathbb{R}^{4}$ with equation $x+y+z+w=10$

Describe the plane that contains this point and line.

Intersect these planes.

Intersect each pair, if possible.

• $\{\hskip 1.00006pt\begin{pmatrix}1\\ 1\\ 2\end{pmatrix}+t\begin{pmatrix}0\\ 1\\ 1\end{pmatrix}\mid t\in\mathbb{R}\hskip 1.00006pt\}$,  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 3\\ -2\end{pmatrix}+s\begin{pmatrix}0\\ 1\\ 2\end{pmatrix}\mid s\in\mathbb{R}\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt\begin{pmatrix}2\\ 0\\ 1\end{pmatrix}+t\begin{pmatrix}1\\ 1\\ -1\end{pmatrix}\mid t\in\mathbb{R}\hskip 1.00006pt\}$,  $\{\hskip 1.00006pts\begin{pmatrix}0\\ 1\\ 2\end{pmatrix}+w\begin{pmatrix}0\\ 4\\ 1\end{pmatrix}\mid s,w\in\mathbb{R}\hskip 1.00006pt\}$

How should we define $\mathbb{R}^{0}$?

Euclid describes a plane as “a surface which lies evenly with the straight lines on itself”. Commentators such as Heron have interpreted this to mean, “(A plane surface is) such that, if a straight line pass through two points on it, the line coincides wholly with it at every spot, all ways”. (Translations from , pp. 171-172.) Do planes, as described in this section, have that property? Does this…