Introduction to Column Vectors Flashcards

All 6 cards in this deck

What does each number in a column vector $\begin{pmatrix}x\\y\end{pmatrix}$ mean?

e.g. $\begin{pmatrix}3\\-2\end{pmatrix}$

Top = movement right (left if negative), bottom = movement up (down if negative).

e.g. 3 right and 2 down.

What are the steps to write a translation as a column vector from a diagram?

e.g. a point is translated 4 left and 1 up

1. Count the movement across: 4 left $= -4$
2. Count the movement up or down: 1 up $= +1$
3. Write across on top, up/down underneath: $\begin{pmatrix}-4\\1\end{pmatrix}$

How do you add or subtract two column vectors?

e.g. $\begin{pmatrix}3\\4\end{pmatrix}+\begin{pmatrix}5\\-2\end{pmatrix}$

Add (or subtract) the top numbers, then the bottom numbers separately.

e.g. $\begin{pmatrix}8\\2\end{pmatrix}$

What are the steps to work out $2\mathbf{a}-3\mathbf{b}$ as a column vector?

e.g. $\mathbf{a}=\begin{pmatrix}3\\4\end{pmatrix}$, $\mathbf{b}=\begin{pmatrix}5\\-2\end{pmatrix}$

1. Multiply each part of $\mathbf{a}$ by 2: $\begin{pmatrix}6\\8\end{pmatrix}$
2. Multiply each part of $\mathbf{b}$ by 3: $\begin{pmatrix}15\\-6\end{pmatrix}$
3. Subtract top from top, bottom from bottom: $\begin{pmatrix}-9\\14\end{pmatrix}$

True or false? To multiply a column vector by 3 you multiply only the top number by 3.

False. You multiply both the top and the bottom number by 3.

How can you tell from their column vectors that two vectors are parallel?

One is a scalar multiple of the other.

e.g. $\begin{pmatrix}6\\9\end{pmatrix}=3\begin{pmatrix}2\\3\end{pmatrix}$, so they are parallel.