A vector contains magnitude and direction
A scalar has magnitude alone, such as a mass of 5 kg or a temperature of 20°C. A vector has magnitude and direction, such as a of 5 m east. In the coordinate plane a column vector [[a], [b]] records horizontal movement a and vertical movement b. We also write (a, b) when the vector context is clear. The vector (3, 4) moves three units right and four up. Its magnitude is √(3² + 4²) = 5.
A position vector starts at the origin and locates a point. If points A and B have position vectors a and b, the from A to B is. With A(1, 2) and B(5, −1), which has magnitude 5. The reverse vector has the same magnitude but opposite direction. A vector is not tied to one starting point: equal vectors represent the same wherever their arrows are drawn.
Add vectors and multiply by scalars
Add corresponding components: (2, 3) + (4. Geometrically, place the second arrow's tail at the first arrow's head; the resultant runs from the original start to the final endpoint. Subtraction a − b is addition of the opposite vector −b. A closed journey has zero resultant even if a substantial distance was travelled. Distinguish total path length, which adds nonnegative lengths, from , which combines directions.
A scalar k multiplies every component. For, and. A positive scalar preserves direction and changes magnitude by its factor; a negative scalar reverses direction. The magnitude of ka is |k| times the magnitude of a. Parallel nonzero vectors are scalar multiples. For example, (6, 9) is parallel to (2, 3), with factor 3. Check both components when deciding parallelism; matching only one ratio is insufficient.
Use vectors to divide and prove
The midpoint of position vectors a and b has position vector (a + b)/2. A point P one third of the way from A to B has. This is a controlled from A, not a separate formula to memorise without direction. To prove three points collinear, show that two vectors from a common point are scalar multiples. If and, then AC = 2.5AB, so the points lie on one line.
A parallelogram provides a useful vector identity. If adjacent sides from A are vectors u and v, the opposite vertex has position a + u + v. Its diagonals have the same midpoint, which can establish a parallelogram from coordinates. Vector arguments should name the arrows: the vector AB is not the same as a point B or the length AB. In exact work, retain vector notation until a magnitude or coordinate is explicitly requested.
Translate and reflect a figure
A translation by (a, b) maps every point (x, y) to (x + a, y + b). It preserves lengths, angles and orientation. A complete description gives the translation vector. A reflection needs its mirror line. Reflection in the x-axis maps (x, y) to (x, −y); reflection in the y-axis maps it to (−x, y). Reflection in interchanges coordinates, giving (y, x), while reflection in gives (−y, −x). Plot a few corresponding points to check the rule.
Reflections preserve shape and size but reverse orientation. A point on the mirror line remains fixed, and the segment joining a point to its image is perpendicular to the mirror line and bisected by it. These geometric facts help identify a mirror line from a pair of figures. A rotation also preserves size, but a complete description needs centre, angle and direction. “Rotate by 90°” is incomplete unless the centre and clockwise or anticlockwise direction are specified.
Rotate and enlarge
About the origin, a 90° anticlockwise rotation maps (x, y) to (−y, x), a 90° clockwise rotation to (y, −x), and a 180° rotation to (−x, −y). For a different centre C, first express the point relative to C, apply the rotation, then add the centre back. A point can be used to verify direction: rotating (1, 0) 90° anticlockwise should give (0, 1). This simple check prevents exchanging the two 90° rules.
An enlargement of scale factor k and centre C sends position p to c + k(p − c). With centre the origin, (x, y) maps to (kx, ky). Positive k greater than 1 enlarges; 0 < k < 1 reduces. A negative factor puts the image on the opposite side of the centre. Lengths scale by |k|, areas by k², and orientation effects depend on the transformation. State the centre and signed scale factor, not just the word “enlargement”.
Matrices and the order of transformations
Origin-fixing linear transformations can be represented by 2 × 2 matrices multiplying coordinate columns. Reflection in the x-axis uses [[1, 0], [0, −1]]. Rotation 90° anticlockwise uses [[0, −1], [1, 0]]. The columns are the images of the basis vectors (1, 0) and (0, 1). Ordinary translations are not represented by a 2 × 2 matrix alone because such multiplication leaves the origin fixed. Treat the added translation separately at this level.
When matrix M acts first and N second, the combined matrix is NM, since. Order generally matters. Reflecting then rotating can differ from rotating then reflecting, just as matrix multiplication can be noncommutative. Self-check: find AB for A(−1, 3), B(2, 7), and rotate (2, 5) 90° clockwise about the origin. The answers are (3, 4), magnitude 5, and (5, −2). Finish by checking invariants: a reflection or rotation should preserve distances, while an enlargement should change them by the specified factor.
Describe invariant points and area changes
A point is invariant under a transformation if it stays in its original position. Every point on a reflection's mirror line is invariant. A nontrivial rotation fixes its centre; an enlargement with factor other than 1 fixes its centre. A nonzero translation fixes no point. These observations can help identify a transformation from a diagram, but a full description still needs its defining data.
For an origin-fixing matrix transformation in the plane, the absolute value of its determinant is the area scale factor. The matrix [[2, 0], [0, 3]] stretches horizontal lengths by 2 and vertical lengths by 3, so areas multiply by 6. It does not preserve shape in general because the two directions have different length factors. A negative determinant indicates orientation reversal, while its magnitude still gives the nonnegative area factor.