Flashcards · 0580 · Topic 7

Transformations and vectors flashcards

25 cards on Topic 7, subtopics 7.1 to 7.4. Pick a subtopic or study them all.

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What must you state to describe each transformation fully?: Reflection: the mirror line. Rotation: angle, direction (unless 180°) and centre. Enlargement: scale factor and centre. Translation: the column vector.
Image of (x, y) after a reflection in the x-axis and in the y-axis: x-axis (y = 0): (x, −y) y-axis (x = 0): (−x, y)
Image of (x, y) after a rotation about the origin of 90° clockwise, 90° anticlockwise and 180°: 90° clockwise: (y, −x) 90° anticlockwise: (−y, x) 180°: (−x, −y)
Enlarge the point (2, 5) by scale factor 3 about the origin: (6, 15): multiply both coordinates by the scale factor (never add).
A translation by the column vector with top entry 4 and bottom entry −2: Move 4 units in the positive x-direction (right) and 2 units in the negative y-direction (down).
Which transformations give an image congruent to the object?: Reflection, rotation and translation. An enlargement (scale factor not 1 or −1) gives a similar image.
Image of (x, y) after a reflection in y = x and in y = −xExtended: y = x: (y, x) y = −x: (−y, −x)
Enlargement with a negative scale factor: enlarge (3, 1) by scale factor −2 about the originExtended: (−6, −2). The image is on the opposite side of the centre and is turned upside down.
Which single enlargement has the same effect as a rotation of 180° about the origin?Extended: An enlargement with scale factor −1, centre the origin.

Vector and scalarExtended: A vector has magnitude and direction (e.g. a displacement). A scalar has magnitude only (e.g. a distance). A vector is printed as AB with an arrow or as bold a; by hand, underline the letter.
a = (3, 4) and b = (−6, 1). Find a + b and 2a − b.Extended: a + b = (−3, 5) 2a − b = (6 − (−6), 8 − 1) = (12, 7)
A is (2, 5) and B is (7, 1). Find the column vector AB.Extended: End minus start: (7 − 2, 1 − 5) = (5, −4).
Effect of multiplying a vector by a scalar kExtended: k > 1: same direction, longer. 0 < k < 1: same direction, shorter. k < 0: opposite direction, length × |k|.
How are AB and BA related?Extended: BA = −AB: the same length, in the opposite direction.

Magnitude of the vector (x, y)Extended: |v| = √(x² + y²): Pythagoras on the components.
Find the magnitude of (−5, 12)Extended: √(25 + 144) = √169 = 13
Find the exact magnitude of (4, −6)Extended: √(16 + 36) = √52 = 2√13
The vector (k, 6) has magnitude 10. Find k.Extended: k² + 36 = 100, so k² = 64 and k = 8 or k = −8.
Is |a + b| the same as |a| + |b|?Extended: Not usually. Find the vector a + b first, then its magnitude. They are equal only when a and b point in the same direction.

Position vectorExtended: The vector from the origin O to a point: the position vector of A is OA.
OA = a and OB = b. Write AB in terms of a and b.Extended: AB = AO + OB = −a + b = b − a
OA = a, OB = b and M is the midpoint of AB. Find OM.Extended: OM = OA + ½AB = a + ½(b − a) = ½a + ½b
OA = a, OB = b. P lies on AB with AP : PB = 1 : 2. Find OP.Extended: AP = ⅓AB = ⅓(b − a) OP = a + ⅓(b − a) = ⅔a + ⅓b
How do you show two vectors are parallel?Extended: Show that one is a scalar multiple of the other, e.g. 4a − 2b = 2(2a − b).
How do you show that points A, B and C are collinear?Extended: Show that AB is a scalar multiple of AC (or BC). Parallel vectors that share a point lie on the same straight line.