Transformations, Matrices And Vectors
Visão Geral
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
Este tema esta mapeado para mostrar como as ideias se ligam.
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Nota de Aula
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Avaliação da Lição
Parabéns por concluir a lição em Transformations, Matrices And Vectors. Agora que você explorou o conceitos e ideias-chave, é hora de colocar seu conhecimento à prova. Esta seção oferece uma variedade de práticas perguntas destinadas a reforçar sua compreensão e ajudá-lo a avaliar sua compreensão do material.
Irá encontrar uma mistura de tipos de perguntas, incluindo perguntas de escolha múltipla, perguntas de resposta curta e perguntas de redação. Cada pergunta é cuidadosamente elaborada para avaliar diferentes aspetos do seu conhecimento e competências de pensamento crítico.
Use esta secção de avaliação como uma oportunidade para reforçar a tua compreensão do tema e identificar quaisquer áreas onde possas precisar de estudo adicional. Não te deixes desencorajar pelos desafios que encontrares; em vez disso, vê-os como oportunidades de crescimento e melhoria.
- 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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