Flashcards · 4024 · Topic 7

Transformations and vectors flashcards

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

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Describe a transformation fully: what must you state for each type?: Reflection: the mirror line as an equation. Rotation: angle, direction and centre. Enlargement: scale factor and centre. Translation: the column vector.
Reflection of (x, y) in y = x and in y = −x: In y = x: (y, x). In y = −x: (−y, −x).
Rotation of (x, y) through 90° about the origin: Anticlockwise: (−y, x). Clockwise: (y, −x). Half turn (180°): (−x, −y).
Negative scale factor of enlargement: what does it do?: The image is on the opposite side of the centre and is inverted. Lengths are multiplied by the size of k. A scale factor of −1 is the same as a 180° rotation about the centre.
A shape of area 20 cm² is enlarged by scale factor 3/2. Find the image area.: Areas are multiplied by k² = 9/4. 20 × 9/4 = 45 cm².
P(3, −1) is translated to P′(−2, 5). Find the translation vector.: Image minus object: x: −2 − 3 = −5, y: 5 − (−1) = 6. Column vector with top −5 and bottom 6.

Vector: A quantity with size (magnitude) and direction.
Scalar: An ordinary number used to multiply a vector.
Position vector: The vector from the origin O to a point: OA = a.
p = (1, 4) and q = (−3, 2) as column vectors. Find 2p + 3q.: 2p = (2, 8), 3q = (−9, 6). 2p + 3q = (−7, 14).
OA = a and OB = b. Write AB and BA.: AB = OB − OA = b − a. BA = −AB = a − b.

Magnitude of the column vector (x, y): |(x, y)| = √(x² + y²), the length of the vector. It is always a non-negative number.
Find |(8, 15)|: √(64 + 225) = √289 = 17
The vector (k, 4) has magnitude 5 and k is positive. Find k.: k² + 16 = 25, so k² = 9 and k = 3.
a = (3, 4) and b = (5, 12). Is |a + b| equal to |a| + |b|?: No. |a| = 5 and |b| = 13, but a + b = (8, 16), so |a + b| = √320 = 8√5 ≈ 17.9, not 18.

Parallel vectors: One is a scalar multiple of the other. Example: 9a − 6b = 1.5(6a − 4b), so they are parallel.
How do you show that three points P, Q and R are collinear?: 1. Find PQ and QR. 2. Show one is a scalar multiple of the other, so they are parallel. 3. State that they share the point Q. 4. Conclude that P, Q and R lie on a straight line.
Position vector of the midpoint M of AB: OM = ½(a + b), where OA = a and OB = b.
AP : PB = 1 : 3. Find OP in terms of a and b.: AP = ¼AB = ¼(b − a). OP = a + ¼(b − a) = ¾a + ¼b.
A point P divides AB in the ratio 2 : 3. What fraction of AB is AP?: 2/5 (the first part out of 2 + 3 = 5 parts), not 2/3.