Vectors Flashcards

All 18 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.

What does the notation $\vec{AB}$ mean?

The vector of the journey from point $A$ to point $B$.

How do you draw a given column vector on a grid, starting from a point $P$?

e.g. draw $\begin{pmatrix}4\\-1\end{pmatrix}$ from $P$

Count the top number across and the bottom number up/down from $P$, then draw an arrow from $P$ to that point.

e.g. 4 right and 1 down from $P$, arrow pointing away from $P$.

On a diagram, how does $-\mathbf{a}$ compare with $\mathbf{a}$?

Same length and parallel, but the arrow points in the opposite direction.

On a diagram, how does $2\mathbf{d}$ compare with $\mathbf{d}$?

Twice as long, in the same direction.

True or false? Two arrows drawn in different places on a grid can represent the same vector.

True. If they have the same length and the same direction they are the same vector, wherever they are drawn.

What are the steps to draw $\mathbf{a}+\mathbf{b}$ on a grid?

e.g. $\mathbf{a}=\begin{pmatrix}2\\1\end{pmatrix}$, $\mathbf{b}=\begin{pmatrix}1\\3\end{pmatrix}$

1. Draw $\mathbf{a}$ from the starting point: 2 right, 1 up
2. Draw $\mathbf{b}$ starting at the tip of $\mathbf{a}$: 1 right, 3 up
3. Draw and label $\mathbf{a}+\mathbf{b}$ as the arrow from the start to the final point: $\begin{pmatrix}3\\4\end{pmatrix}$

If $\vec{OA}=\mathbf{a}$ and $\vec{OB}=\mathbf{b}$, what is $\vec{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$?

$\mathbf{b}-\mathbf{a}$

(Go backwards along $\mathbf{a}$ to $O$, then forwards along $\mathbf{b}$.)

How is $\vec{BA}$ related to $\vec{AB}$?

$\vec{BA}=-\vec{AB}$ — same length, opposite direction.

What are the steps to find a vector path in terms of $\mathbf{a}$ and $\mathbf{b}$?

e.g. find $\vec{AD}$ where $\vec{OA}=\mathbf{a}$ and $\vec{OD}=-\mathbf{b}$

1. Choose a route along known vectors: $A\to O\to D$
2. Write each step, using a minus sign when travelling backwards along a vector: $-\mathbf{a}$ then $-\mathbf{b}$
3. Add the steps: $\vec{AD}=-\mathbf{a}-\mathbf{b}$

$M$ is the midpoint of $AB$ and $\vec{AB}=\mathbf{b}$. What is $\vec{AM}$?

$\frac{1}{2}\mathbf{b}$

True or false? If $M$ is the midpoint of $AB$ and $\vec{AB}=2\mathbf{a}$, then $\vec{MB}=\mathbf{a}$.

True. $\vec{AM}$ and $\vec{MB}$ are each half of $\vec{AB}$.

How can you tell that two vectors written in terms of $\mathbf{a}$ and $\mathbf{b}$ are parallel?

e.g. $\mathbf{a}+\mathbf{b}$ and $2\mathbf{a}+2\mathbf{b}$

One can be written as a scalar multiple of the other.

e.g. $2\mathbf{a}+2\mathbf{b}=2(\mathbf{a}+\mathbf{b})$, so they are parallel.