LEARNING OBJECTIVES
By the end of this chapter, you should be able to:
- recognise and describe four transformations
- combine transformations
- use positive, fractional and negative enlargement factors
- add, subtract and scale vectors
- calculate vector magnitudes
- use position vectors in geometrical proofs
THE BIG IDEA
Describe movement and scale precisely, then use directed quantities to prove geometrical relationships.
A transformation maps every point of a shape to a new position. A vector records a movement with both magnitude and direction.
Precise descriptions matter: naming only rotation or enlargement is incomplete without its centre and other defining information.
SECTION 01
The four transformations
A reflection needs a mirror line. A rotation needs a centre, angle and direction. An enlargement needs a centre and scale factor. A translation needs a column vector.
A negative enlargement scale factor places the image on the opposite side of the centre. A fractional factor produces a smaller image.
KEY IDEAS
- Reflection, rotation and translation preserve lengths and angles.
- Enlargement preserves angles and multiplies lengths by |scale factor|; a negative factor reverses the direction through the centre.
- A combination applies transformations in the stated order, using each new image as the input to the next transformation.
Describing a rotation
- Join a point and its image to a possible centre.
- Repeat with another pair; perpendicular bisectors meet at the centre.
- Measure the angle and decide clockwise or anticlockwise.
Answer: A complete answer has the form rotation 90° clockwise about (2, −1).
SECTION 02
Vector arithmetic
Vectors add head-to-tail. Subtracting b means adding −b, which has the same magnitude as b but opposite direction.
A scalar changes a vector's magnitude and may reverse its direction. Equal vectors have equal magnitude and direction even if drawn in different places.
Combining column vectors
- a = (4, −1) and b = (−2, 5).
- 2a = (8, −2).
- 2a − b = (8, −2) − (−2, 5).
Answer: 2a − b = (10, −7).
SECTION 03
Magnitude and direction
The magnitude of a vector is its length. Its horizontal and vertical components form a right-angled triangle, so Pythagoras gives the formula.
A zero vector has magnitude zero and no defined direction.
Magnitude of vector (−7, 24)
- Square both components: (−7)² = 49 and 24² = 576.
- Add: 49 + 576 = 625.
- Take the square root.
Answer: The magnitude is 25.
SECTION 04
Vector geometry and proof
Choose two independent vectors and express every required path in terms of them. Different routes between the same points give equal vectors.
If one vector is a scalar multiple of another, they are parallel. If two directed segments meeting at a point are parallel, the three points are collinear.
Section ratios can be handled by taking the appropriate fraction of a whole vector.
Showing a midpoint relationship
- OA = a and OB = b. M is the midpoint of AB.
- AB = b − a.
- AM = 1/2(b − a).
- OM = OA + AM = a + 1/2(b − a).
Answer: OM = 1/2(a + b), so the midpoint position vector is the average of the endpoints.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Every transformation needs its defining details.
- Negative enlargement reverses direction through the centre.
- Vectors add by components or head-to-tail.
- Magnitude uses Pythagoras.
- Position-vector differences describe directed segments.
- Scalar multiples provide evidence for parallel or collinear lines.