Vectors#
Table of Contents#
\(p = (2, 3)\) is a point but \(\mathbf{p} = \begin{bmatrix}2\\3\\\end{bmatrix}\) (a \(2 \times 1\) matrix) is a vector in \(\mathbb{R}^2\) (two-dimensional space)
\(p = (3, 7, 5)\) is a point but \(\mathbf{p} = \begin{bmatrix}3\\7\\5\\\end{bmatrix}\) (a \(3 \times 1\) matrix) is a vector in \(\mathbb{R}^3\) (three dimensional space)
[ Definition ] Vector Equality
Two vectors are equal iff their corresponding entries are equal.
\(\begin{bmatrix}2\\3\\\end{bmatrix} \ne \begin{bmatrix}3\\2\\\end{bmatrix}\)
[ Definition ] Vector Addition
\(\begin{bmatrix}v_1\\v_2\\\end{bmatrix} + \begin{bmatrix}w_1\\w_2\\\end{bmatrix} = \begin{bmatrix}v_1+w_1\\v_2+w_2\\\end{bmatrix}\)
Geometrically: parallelogram rule
\( \begin{bmatrix}a\\b\\c\\\end{bmatrix} + \begin{bmatrix}\alpha\\\beta\\\gamma\\\end{bmatrix} = \begin{bmatrix}a + \alpha\\b + \beta\\c + \gamma\\\end{bmatrix} \)
[ Definition ] Multiplication of a scalar with a vector
\(r\begin{bmatrix}v_1\\v_2\\\end{bmatrix} = \begin{bmatrix}rv_1\\rv_2\\\end{bmatrix}\)
\(r\begin{bmatrix}a\\b\\c\\\end{bmatrix} = \begin{bmatrix}ra\\rb\\rc\\\end{bmatrix}\)
EXAMPLE
Let \(\mathbf{u} = \begin{bmatrix}0\\1\\\end{bmatrix}, \mathbf{v} = \begin{bmatrix}3\\2\\\end{bmatrix}, \mathbf{w} = \begin{bmatrix*}[r]2\\-1\\\end{bmatrix*}\). Then the following is true.
\(5\mathbf{u} = 5\begin{bmatrix}0\\1\\\end{bmatrix} = \begin{bmatrix}0\\5\\\end{bmatrix}\)
\(2\mathbf{v} = 2\begin{bmatrix}3\\2\\\end{bmatrix} = \begin{bmatrix}6\\4\\\end{bmatrix}\)
\(-3\mathbf{w} = -3\begin{bmatrix*}[r]2\\-1\\\end{bmatrix*} = \begin{bmatrix*}[r]-6\\3\\\end{bmatrix*}\)
\( 5\mathbf{u} + 2\mathbf{v} - 3\mathbf{w} = \begin{bmatrix}0\\5\\\end{bmatrix} + \begin{bmatrix}6\\4\\\end{bmatrix} + \begin{bmatrix*}[r]-6\\3\\\end{bmatrix*} = \begin{bmatrix*}[r] 0 & + & 6 & + & (-6) \\ 5 & + & 4 & + & 3 \\ \end {bmatrix*} = \begin{bmatrix*}[r] 0 \\ 12 \\ \end {bmatrix*} \)