Column Vectors Flashcards

A column vector is written as two numbers in a vertical format, e.g., \( \begin{pmatrix} a \\ b \end{pmatrix} \), where \(a\) is the horizontal component and \(b\) is the vertical component.

The top number in a column vector represents the movement in the horizontal direction (left or right).

The bottom number in a column vector represents the movement in the vertical direction (up or down).

To add two column vectors, add their corresponding components: \( \begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix} \).

To subtract one column vector from another, subtract their corresponding components: \( \begin{pmatrix} a \\ b \end{pmatrix} - \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a-c \\ b-d \end{pmatrix} \).

To multiply a column vector by a scalar, multiply each component by the scalar: \( k \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix} \).

The magnitude (length) of a vector \( \begin{pmatrix} a \\ b \end{pmatrix} \) is calculated using \( \sqrt{a^2 + b^2} \).

The zero vector is \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \) and represents no movement.

Two vectors are parallel if one is a scalar multiple of the other, e.g., \( \begin{pmatrix} 2 \\ 4 \end{pmatrix} \) and \( \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) are parallel.

The resultant vector is the single vector obtained by adding two or more vectors together.