Flashcards · 11 cards

Vectors & transformations

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Rule
What does the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ mean?
Answer
3 steps across (right) and 2 steps down: the top is across, the bottom is up.
A column vectorTop: steps across. Bottom: steps up
From the lesson: Vectors
Rule
How do you add two vectors?
Answer
Head to tail: add the tops and add the bottoms.
Adding vectorsHead to tail: add the tops, add the bottoms
From the lesson: Vectors
Rule
The magnitude of $\begin{pmatrix} x \\ y \end{pmatrix}$ ?
Answer
$\sqrt{x^2 + y^2}$ : the arrow is the hypotenuse.
MagnitudeThe arrow is the hypotenuse: length √(x² + y²)
From the lesson: Vectors
Rule
$\overrightarrow{AB}$ from the position vectors of $A$ and $B$ ?
Answer
$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$ : end minus start.
From A to BAB = OB − OA: end minus start
From the lesson: Vectors
Rule
When are two vectors parallel?
Answer
When one is a number times the other. $2\mathbf a$ is parallel to $\mathbf a$ and twice as long; $-\mathbf a$ points the other way.
Parallel vectors2a is twice as long as a; −a points the other way
From the lesson: Vectors
Rule
What does a translation by $\begin{pmatrix} a \\ b \end{pmatrix}$ do?
Answer
It moves every point by the same vector: $(x, y) \to (x + a, y + b)$ .
TranslationEvery point moves by the same vector
From the lesson: Transformations
Know it
Reflections in the $x$ -axis and in the $y$ -axis?
Answer
$x$ -axis: $(x, y) \to (x, -y)$ . $y$ -axis: $(x, y) \to (-x, y)$ .
From the lesson: Transformations
Rule
Reflections in $y = x$ and in $y = -x$ ?
Answer
$y = x$ : $(x, y) \to (y, x)$ . $y = -x$ : $(x, y) \to (-y, -x)$ .
Reflection in y = xThe coordinates swap: (3, 1) goes to (1, 3)
From the lesson: Transformations
Rule
Rotations about the origin through $90^\circ$ anticlockwise, $90^\circ$ clockwise and $180^\circ$ ?
Answer
Anticlockwise: $(x, y) \to (-y, x)$ . Clockwise: $(x, y) \to (y, -x)$ . $180^\circ$ : $(x, y) \to (-x, -y)$ . Check with $(1, 0)$ : anticlockwise sends it to $(0, 1)$ .
Rotation 90° anticlockwise(3, 1) goes to (−1, 3): swap, then change the sign of the new x
From the lesson: Transformations
Rule
An enlargement from $O$ with scale factor $k$ ?
Answer
$(x, y) \to (kx, ky)$ . A negative $k$ puts the image on the other side of $O$ .
Enlargement from OScale factor 2 (teal) and −½ (orange): lines from O pass through matching points
From the lesson: Transformations
Which method?
WAEC 2018 · Paper 2 · Q13 (a)
The points $R(3, -6)$ , $S(6, -2)$ and $T(p, q)$ are on the $xy$ -plane. If $\frac13\overrightarrow{OR} + \overrightarrow{OS} + \overrightarrow{OT} = \overrightarrow{RS}$ , find the coordinates of $T$ .
How do you turn the vector equation into numbers?
Answer
Write each vector as a column: $\overrightarrow{OR} = \begin{pmatrix} 3 \\ -6 \end{pmatrix}$ , and $\overrightarrow{RS} = \overrightarrow{OS} - \overrightarrow{OR}$ . Then match the tops and the bottoms to find $p$ and $q$ .
From the lesson: Vectors Try the question