ZIMSEC O Level · Mathematics · Form 3, Form 4
Vectors
A vector has size (magnitude) and direction. It is written as a column (x over y) or as AB with an arrow.
- Add vectors by adding their components.
- The magnitude of the vector (x, y) is √(x² + y²).
- AB = OB − OA (position vectors).
Worked example: A(2, 1) and B(5, 5). Find vector AB and its magnitude.
AB = (5 − 2, 5 − 1) = (3, 4). |AB| = √(9 + 16) = 5 units
Matrices
- Add or subtract matrices of the same order, element by element.
- Multiply rows by columns. A 2×2 matrix times a 2×2 matrix gives a 2×2 matrix.
- For M = (a b; c d), the determinant is ad − bc and the inverse is 1/(ad − bc) × (d −b; −c a).
Worked example: Find the determinant and inverse of M = (4 3; 2 2).
det M = 4×2 − 3×2 = 2. M⁻¹ = ½ × (2 −3; −2 4) = (1 −1.5; −1 2)
Transformations
- Translation: moves a shape by a vector.
- Reflection: in a mirror line, e.g. y = x.
- Rotation: needs a centre, an angle and a direction.
- Enlargement: needs a centre and a scale factor k. Area scale factor = k².
Some transformations can be done with matrices, e.g. (0 −1; 1 0) is a rotation of 90° anticlockwise about the origin.
Exam tip: To describe a transformation fully, give ALL its details – a rotation with no centre loses marks.
Check yourself
- p = (3, −2) and q = (−1, 4). Find 2p + q.
- Find the determinant of (5 2; 3 1).
- A shape of area 6 cm² is enlarged by scale factor 3. Find the area of the image.
Show answers
- (5, 0)
- 5 − 6 = −1
- 54 cm²
Questions about this topic (0)
No questions about this topic yet.
Sign in or create a free account to join in.