Vectors Flashcards

All 29 cards in this deck

What does the column vector $\begin{pmatrix}x\\y\end{pmatrix}$ describe?

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

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

e.g. 5 right and 2 down.

What are the steps to add or subtract two column vectors?

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

1. Combine the top numbers: $3+5=8$
2. Combine the bottom numbers: $4+(-2)=2$
3. Write as one column vector: $\begin{pmatrix}8\\2\end{pmatrix}$

How do you multiply a column vector by a scalar?

e.g. $3\begin{pmatrix}2\\-1\end{pmatrix}$

Multiply each component by the scalar.

e.g. $\begin{pmatrix}6\\-3\end{pmatrix}$

True or false? $-\mathbf{a}$ has the same length as $\mathbf{a}$ but points in the opposite direction.

True. Negating a vector reverses its direction and leaves its length unchanged.

How is a vector written in print, and how should you write it by hand?

Print: bold lower case, $\mathbf{a}$. By hand: underlined, $\underline{a}$.

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

e.g. $\mathbf{a}=\begin{pmatrix}3\\2\end{pmatrix}$, $\mathbf{b}=\begin{pmatrix}-1\\4\end{pmatrix}$

1. Draw $\mathbf{a}$: 3 right, 2 up
2. From the tip of $\mathbf{a}$, draw $\mathbf{b}$: 1 left, 4 up
3. Draw the resultant from the start of $\mathbf{a}$ to the end of $\mathbf{b}$: $\begin{pmatrix}2\\6\end{pmatrix}$

What must a vector drawn on a grid always show?

An arrow giving its direction (plus its label if asked for).

How do you draw $-2\mathbf{a}$ given $\mathbf{a}$ on a grid?

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

Twice as long as $\mathbf{a}$ and in the opposite direction.

e.g. draw $\begin{pmatrix}-2\\-6\end{pmatrix}$

On a diagram, what path represents $\mathbf{a}-\mathbf{b}$?

Travel along $\mathbf{a}$, then along $\mathbf{b}$ backwards (against its arrow).

What does the notation $|\mathbf{a}|$ mean?

The magnitude (length) of the vector $\mathbf{a}$.

What is the formula for the magnitude of the vector $\begin{pmatrix}x\\y\end{pmatrix}$?

$\sqrt{x^2+y^2}$

What are the steps to find the length of a column vector?

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

1. Square each component: $9$ and $16$
2. Add them: $25$
3. Take the square root: $5$

True or false? $\begin{pmatrix}-3\\4\end{pmatrix}$ and $\begin{pmatrix}3\\-4\end{pmatrix}$ have the same magnitude.

True. Squaring removes the signs, so only the directions differ.

If a vector is multiplied by the scalar $k$, what happens to its length?

The length is multiplied by $|k|$.

e.g. $|3\mathbf{a}|=3|\mathbf{a}|$

What is a position vector?

The vector from the origin $O$ to a point.

e.g. $A(3,5)$ has position vector $\begin{pmatrix}3\\5\end{pmatrix}$.

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

The vector (displacement) from point $A$ to point $B$.

If $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$, what is $\overrightarrow{AB}$?

$\mathbf{b}-\mathbf{a}$ (end minus start).

True or false? $\overrightarrow{AB}=\overrightarrow{BA}$

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

What are the steps to find $\overrightarrow{PQ}$ from the coordinates of $P$ and $Q$?

e.g. $P(1,2)$ and $Q(4,7)$

1. Write the position vectors: $\mathbf{p}=\begin{pmatrix}1\\2\end{pmatrix}$, $\mathbf{q}=\begin{pmatrix}4\\7\end{pmatrix}$
2. Subtract: $\mathbf{q}-\mathbf{p}$
3. Answer: $\begin{pmatrix}3\\5\end{pmatrix}$

What are the steps to express a vector such as $\overrightarrow{FE}$ in terms of $\mathbf{a}$ and $\mathbf{b}$?

e.g. $\overrightarrow{FC}=\mathbf{a}$, $\overrightarrow{CD}=-\mathbf{b}$, $\overrightarrow{DE}=\mathbf{a}+\mathbf{b}$

1. Find a route from $F$ to $E$ along known vectors: $F\to C\to D\to E$
2. Add them in order, using $-$ for any travelled backwards: $\mathbf{a}-\mathbf{b}+\mathbf{a}+\mathbf{b}$
3. Simplify: $2\mathbf{a}$

What does the triangle law give for $\overrightarrow{PQ}+\overrightarrow{QR}$?

$\overrightarrow{PR}$

In parallelogram $OABC$ with $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$, what is $\overrightarrow{AB}$?

$\mathbf{c}$ — opposite sides of a parallelogram are equal vectors.

True or false? Two different routes between the same two points give different vector expressions.

False. Every route simplifies to the same vector.

What does 'give your answer in its simplest form' mean for a vector expression?

Collect like terms in $\mathbf{a}$ and $\mathbf{b}$.

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

If $M$ is the midpoint of $AB$, what is $\overrightarrow{AM}$ in terms of $\overrightarrow{AB}$?

$\frac{1}{2}\overrightarrow{AB}$

$M$ lies on $OR$ with $OM:MR=2:3$. What is $\overrightarrow{OM}$ in terms of $\overrightarrow{OR}$?

$\frac{2}{5}\overrightarrow{OR}$ — $M$ is 2 parts out of 5 along $OR$.

What are the steps to prove that points $A$, $B$ and $C$ are collinear (lie on a straight line)?

e.g. $\overrightarrow{AB}=3\mathbf{a}+4\mathbf{b}$, $\overrightarrow{AC}=15\mathbf{a}+20\mathbf{b}$

1. Write both vectors in terms of $\mathbf{a}$ and $\mathbf{b}$
2. Show one is a multiple of the other: $\overrightarrow{AC}=5\overrightarrow{AB}$
3. State they are parallel and share the point $A$, so $A$, $B$, $C$ are collinear

True or false? Showing that $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are multiples of each other proves $A$, $B$, $C$ and $D$ lie on a straight line.

False. It only proves the lines are parallel; collinearity also needs a common point.

$D$, $E$, $F$ are on a straight line and $\overrightarrow{DF}=-2.5\overrightarrow{DE}$. What is the ratio of lengths $DF:DE$?

$5:2$ — use the size of the scalar and ignore its sign.