Vectors and Transformations
Vectors and transformations
Vector work includes column vectors, magnitude, addition, subtraction, scalar multiplication and position vectors. For points A and B, vector AB = b - a when a and b are position vectors. Transformations include translation, reflection, rotation and enlargement; you must describe a transformation fully and perform it on a shape. A translation is described by a column vector, a reflection by its mirror line, a rotation by centre, angle and direction, and an enlargement by centre and scale factor.
Common traps are reversing the order in vector subtraction, forgetting the centre of rotation or enlargement, and confusing clockwise with anticlockwise. For enlargement with negative scale factor, the image lies on the opposite side of the centre. If the scale factor is k, the area of the image is k^2 times the original area. For reflection, the mirror line is the perpendicular bisector of the segment joining each point to its image. Read transformation questions carefully: describe means give all defining information.
Worked example: A(1, 2) and B(4, 6). Vector AB = (4 - 1, 6 - 2) = (3, 4). Its magnitude is sqrt(3^2 + 4^2) = sqrt(25) = 5. An enlargement of this segment by scale factor 2 gives an image length of 10.
Rule of thumb: for vectors, think destination minus start; for transformations, identify the image of one key point and work backwards to the centre or mirror line.