NSW Syllabuses

# Mathematics Extension 2 Stage 6 - Year 12 Extension 2 - Vectors MEX-V1 Further Work with Vectors

## Outcomes

#### A student:

• MEX12-3

uses vectors to model and solve problems in two and three dimensions

• MEX12-7

applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems

• MEX12-8

communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument

## Subtopic Focus

The principal focus of this subtopic is to explore graphical and algebraic representations of quantities with magnitude and direction in two and three dimensions.

Students develop an understanding of connections between the behaviour of lines or quantities and their representation as vectors, and solve problems using geometric approaches.

## Content

• V1.1: Introduction to three dimensional vectors
• Students:
• review the use of two dimensional vectors to model and solve problems
• understand and use a variety of notations and representations for vectors in three dimensions (ACMSM014, ACMSM015, ACMSM016, ACMSM101)
• define the standard unit vector $$\underset{^\sim}{k}$$
• express and use a vector in three dimensions in a variety of forms, including component form and column vector notation
• use column vector notation and ordered triples notation to represent a vector
• perform addition and subtraction of three dimensional vectors and multiplication of three dimensional vectors by a scalar algebraically and geometrically, and interpret these operations in geometric terms
• V1.2: Further operations with three dimensional vectors
• Students:
• define, calculate and use the magnitude of a vector in three dimensions (ACMSM011)
• prove that the magnitude of a vector in three dimensions can be found using: $$\lvert x\underset{^\sim}i +y\underset{^\sim} j + z\underset{^\sim}k\rvert = \sqrt{x^2 + y^2 + z^2}$$
• define and apply the scalar product $$\underset{^\sim}u \cdot \underset{^\sim}v$$ to vectors expressed in component form, where $$\underset{^\sim}u \cdot \underset{^\sim}v = x_1 y_1 + x_2 y_2 + x_3 y_3 = \displaystyle{\sum_{i=1}^3}{x_iy_i}$$, $$\underset{^\sim}{u} = x_1 \underset{^\sim}i + x_2 \underset{^\sim}j + x_3 \underset{^\sim}k$$ and $$\underset{^\sim}v = y_1 \underset{^\sim}i + y_2 \underset{^\sim}j + y_3 \underset{^\sim}k$$
• extend the formula $$\underset{^\sim}u \cdot \underset{^\sim}v = \lvert \underset{^\sim}u \rvert \lvert\underset{^\sim}v \rvert \cos\theta$$ for three dimensions and use to solve problems
• prove geometric results in the plane and construct proofs involving vectors in two and three dimensions (ACMSM102)
• V1.3: Vectors and Vector equations of lines
• Students:
• use Cartesian coordinates in two and three-dimensional space, including plotting points and the equations of spheres (ACMSM103)
• use vector equations of curves in two or three dimensions involving a parametric variable, and determine a corresponding Cartesian equation in the two-dimensional case, where possible (ACMSM104)
• understand and use the vector form of a straight line $$\underset{^\sim}r = \underset{^\sim} a + \lambda \underset{^\sim} b$$, in two dimensions
• understand the significance of all elements used when the equation of a straight line is expressed in the form $$\underset{^\sim}r = \underset{^\sim} a + \lambda \underset{^\sim} b$$
• make connections in two dimensions between the equation $$\underset{^\sim}r = \underset{^\sim} a + \lambda \underset{^\sim} b$$, and $$y = mx + c$$
• determine a vector equation of a straight line or straight-line segment, given the position of two points or equivalent information, in two and three dimensions (ACMSM105)
• determine when two lines in vector form are parallel or perpendicular
• determine when a given point lies on a given line