What is a parallelogram in three dimensions? Recall that vectors are equal if they form opposite sides of a parallelogram. The notion of Equality of vectors a parallelogram in three dimensions requires a discussion of when the opposite sides of a Coordinate systems and position vectors; quadrilateral are coplanar so that it is not components twisted; the simplest criterion is that its diagonals intersect. Applications (e.g. navigation and force) as encountered in Stage 1 Mathematics can readily be extended to three dimensions. Operations with vectors Contexts and applications as mentioned above. Addition of vectors, multiplication by scalars, length of vectors: both geometrically and algebraically Parallel vectors, collinearity, ratio of division of a Ratio of division is needed in the treatment of line segment (internal and external) Bézier curves (see Topic 4: Calculus). Can vectors be multiplied together? These two operations on pairs of vectors provide important geometric information. The What is the meaning of the product? scalar and vector products should be treated in terms of coordinates and of length and angle. Scalar (dot) product and vector (cross) product: Conditions for perpendicularity and parallelism, their properties and their interpretation in and construction of perpendiculars. context Cross-product calculation using the determinant Though determinants may not have been encountered previously, this is a suitable place for their introduction, since they arise naturally in the formulae for area and volume. (Note that the area of a parallelogram in two dimensions can be expressed as a determinant in terms of the coordinates of the defining vectors.) (Determinants are treated in Subtopic 3.4 of Stage 2 Mathematical Studies.) The cross-product of two vectors and in three dimensions is a vector, mutually perpendicular to and to whose length is the area of the parallelogram determined by and The hand rule determines its sense. The components and the vector itself may be expressed using determinants. (Applications of scalar and vector products include work and moments.) When and how is the scalar triple product used? Scalar triple product as volume of a parallelepiped, again using determinant Volume of a parallelepiped notation; test for coplanarity. Test for coplanarity Subtopic 3.2: Lines and Planes Key Questions and Key Ideas Considerations for Developing Teaching and Learning Strategies
What is a parallelogram in three dimensions? Equality of vectors Coordinate systems and position vectors; components Recall that vectors are equal if they form opposite sides of a parallelogram. The notion of a parallelogram in three dimensions requires a discussion of when the opposite sides of a quadrilateral are coplanar so that it is not twisted; the simplest criterion is that its diagonals intersect. Applications (e.g. navigation and force) as encountered in Stage 1 Mathematics can readily be extended to three dimensions. Operations with vectors Addition of vectors, multiplication by scalars, length of vectors: both geometrically and algebraically Contexts and applications as mentioned above. Parallel vectors, collinearity, ratio of division of a line segment (internal and external) Can vectors be multiplied together? What is the meaning of the product? Scalar (dot) product and vector (cross) product: their properties and their interpretation in context Ratio of division is needed in the treatment of Bézier curves (see Topic 4: Calculus). These two operations on pairs of vectors provide important geometric information. The scalar and vector products should be treated in terms of coordinates and of length and angle. Conditions for perpendicularity and parallelism, and construction of perpendiculars. Cross-product calculation using the determinant Though determinants may not have been encountered previously, this is a suitable place for their introduction, since they arise naturally in the formulae for area and volume. (Note that the area of a parallelogram in two dimensions can be expressed as a determinant in terms of the coordinates of the defining vectors.) (Determinants are treated in Subtopic 3.4 of Stage 2 Mathematical Studies.) The cross-product of two vectors and in three dimensions is a vector, mutually perpendicular to and to whose length is the area of the parallelogram determined by and The rule determines its sense. The components and the vector itself may be expressed using determinants. (Applications of scalar and vector products include work and moments.) When and how is the scalar triple product used? Volume of a parallelepiped Test for coplanarity Scalar triple product as volume of a parallelepiped, again using determinant notation; test for coplanarity. Subtopic 3.2: Lines and Planes Key Questions and Key Ideas Considerations for Developing Teaching and Learning Strategies hand