Transformations, Matrices And Vectors
Resumen
Only one of the seven statements in this topic is Core. The other six are Extension Tier only, and they contain the two most distinctive pieces of content on the whole of 9260: vector methods for proving geometrical results, and matrices used to represent transformations of the plane. If you are entered for Papers 1E and 2E this is a topic worth serious time, because almost nobody arrives at it already fluent.
The Core statement asks you to describe and transform two dimensional shapes using single rotations, reflections, translations or enlargements by a positive scale factor, and to distinguish the properties that are preserved under each. On the Extension Tier you add combined transformations and enlargements by fractional and negative scale factors, vector notation with the sum, difference and scalar multiple of vectors and their commutative and associative properties, matrix multiplication and the identity matrix, transformations of the unit square represented by a two by two matrix, and combinations of transformations carried out by multiplying matrices.
Objetivos
- [Core] describe and transform 2D shapes using single rotations, reflections, translations, or enlargements by a positive scale factor and distinguish properties that are preserved under particular transformations
- [Extension] including combined transformations and enlargements by fractional and negative scale factors (Notes: translations will be specified by a vector)
- [Extension] understand and use vector notation; calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector; understand and use the commutative and associative properties of vector addition; solve simple geometrical problems in 2D using vector methods
- [Extension] multiplications of matrices (Notes: multiplying a 2 × 2 matrix by a 2 × 2 matrix or by a 2 × 1 matrix, multiplication by a scalar)
- [Extension] the identity matrix, I (Notes: 2 × 2 only)
- [Extension] transformations of the unit square in the x - y plane (Notes: representation by a 2 × 2 matrix transformations restricted to rotations of 90°, 180° or 270° about the origin, reflections in a line through the origin (ie x = 0, y = 0, y = x, y = -x) and enlargements centred on the origin)
- [Extension] combination of transformations (Notes: using matrix multiplications use of i and j notation is not required)
Mapa mental
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Nota de la lección
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Evaluación de la lección
Felicitaciones por completar la lección del Transformations, Matrices And Vectors. Ahora que has explorado el conceptos e ideas clave, es hora de poner a prueba tus conocimientos. Esta sección ofrece una variedad de prácticas Preguntas diseñadas para reforzar su comprensión y ayudarle a evaluar su comprensión del material.
Te encontrarás con una variedad de tipos de preguntas, incluyendo preguntas de opción múltiple, preguntas de respuesta corta y preguntas de ensayo. Cada pregunta está cuidadosamente diseñada para evaluar diferentes aspectos de tu conocimiento y habilidades de pensamiento crítico.
Utiliza esta sección de evaluación como una oportunidad para reforzar tu comprensión del tema e identificar cualquier área en la que puedas necesitar un estudio adicional. No te desanimes por los desafíos que encuentres; en su lugar, míralos como oportunidades para el crecimiento y la mejora.
- Vector a has components 4 and -2, and vector b has components -3 and 3. What is a - b? A. components 1 and 1 B. components 7 and -5 C. components -7 and 5 D. components 1 and -5 Answer: B
- Which transformation is the only one of the four that reverses the orientation of a shape? A. Translation B. Reflection C. Rotation D. Enlargement with a positive scale factor Answer: B
- Which matrix represents a reflection in the line y = x? A. rows (1, 0) and (0, -1) B. rows (-1, 0) and (0, 1) C. rows (0, 1) and (1, 0) D. rows (0, -1) and (1, 0) Answer: C
- A shape is enlarged with scale factor -3, centre the origin. Which statement is correct? A. The image is smaller and on the same side of the origin B. The image is three times as long and on the opposite side of the origin C. The image is nine times as long and on the same side of the origin D. The transformation is impossible Answer: B
- In the matrix product PQ acting on a column vector, which transformation is applied first? A. P B. Q C. Neither, they act at the same time D. It depends on whether P and Q are reflections Answer: B
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