NSW Syllabuses

# Mathematics Extension 1 Stage 6 - Year 12 Extension 1 - Vectors ME-V1 Introduction to Vectors

## Outcomes

#### A student:

• ME12-2

applies concepts and techniques involving vectors and projectiles to solve problems

• ME12-6

chooses and uses appropriate technology to solve problems in a range of contexts

• ME12-7

evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms

## Subtopic Focus

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

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

## Content

• V1.1: Introduction to Vectors
• Students:
• define a vector as a quantity having both magnitude and direction, and examine examples of vectors, including displacement and velocity (ACMSM010)
• explain the distinction between a position vector and a displacement (relative) vector
• understand and use a variety of notations and representations for vectors in two dimensions (ACMSM014, ACMSM015, ACMSM016)
• use standard notations for vectors: $$\boldsymbol{a}$$, $$\underset{^\sim}a$$ and $$\overrightarrow{AB}$$
• represent vectors graphically in two dimensions as directed line segments
• define unit vectors as vectors of magnitude 1, and the standard unit vectors $$\underset{^\sim}i$$ and $$\underset{^\sim}j$$
• express and use vectors in two dimensions in a variety of forms, including component form, polar form, ordered pairs and column vector notation
• resolve vectors into component form
• perform addition and subtraction of vectors and multiplication of a vector by a scalar algebraically and geometrically, and interpret these operations in geometric terms
• graphically represent a scalar multiple of a vector (ACMSM012)
• use the triangle law and the parallelogram law to find the sum and difference of two vectors (ACMSM013)
• examine and use addition and subtraction of vectors in component form (ACMSM017)
• define and use multiplication by a scalar of a vector in component form (ACMSM018)
• V1.2: Further operations with vectors
• Students:
• define, calculate and use the magnitude of a vector in two dimensions and use the notation $$\lvert \underset{^\sim}u \rvert$$ for the magnitude of a vector $$\underset{^\sim}u$$ (ACMSM011)
• prove that the magnitude of a vector in two dimensions can be found using: $$\lvert x \underset{^\sim}i + y \underset{^\sim}j \rvert = \sqrt { x^2 + y^2 }$$
• identify the magnitude of a displacement vector $$\overrightarrow{AB}$$  as being the distance between the points $$A$$ and $$B$$
• find a unit vector associated with a given non-zero vector, $$\underset{^\sim}u$$
• define and use the direction of a vector in two dimensions (ACMSM011)
• define, calculate and use the scalar (dot) product of two vectors in two dimensions (ACMSM019, ACMSM020)
• demonstrate the equivalence $$\underset{^\sim}u \cdot \!\,\underset{^\sim}v = \lvert \underset{^\sim}u\rvert\lvert \underset{^\sim}v\rvert \cos \theta = x_1 y_1 + x_2 y_2$$ and use this relationship to solve problems
• use the scalar (dot) product $$\underset{^\sim}u \cdot \underset{^\sim}v = \lvert \underset{^\sim}u \rvert \lvert \underset{^\sim}v \rvert \cos\theta$$ where $$\theta$$ is the angle between vectors $$\underset{^\sim}u$$ and $$\underset{^\sim}v$$ to solve problems
• calculate the angle between two vectors using the scalar (dot) product of two vectors in two dimensions
• apply the scalar product, $$\underset{^\sim}u \cdot \underset{^\sim}v$$, to vectors expressed in component form, where $$\underset{^\sim}u \cdot \underset{^\sim}v =x_1 y_1 + x_2 y_2$$
• examine properties of parallel and perpendicular vectors to determine if two vectors are parallel or perpendicular (ACMSM021)
• define and use the projection of one vector onto another (ACMSM022)
• solve problems involving displacement, force and velocity involving vector concepts in two dimensions (ACMSM023)
• prove geometric results and construct proofs involving vectors in two dimensions
• prove that the diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039)
• prove that the midpoints of the sides of a quadrilateral join to form a parallelogram (ACMSM040)
• prove that the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides (ACMSM041)
• determine midpoints and ratios of lengths involving vectors
• V1.3: Projectiles as an application of vectors
• Students:
• model the motion of a projectile as a particle moving with constant acceleration due to gravity and derive the equations of motion of a projectile
• represent the motion of a projectile as a vector with speed and direction
• recognise that the horizontal and vertical motion of a projectile can be presented by horizontal and vertical vectors
• review Newton’s laws of motion as appropriate
• derive the horizontal and vertical equations of motion of a projectile
• understand and explain the limitations of projectile models
• use equations for horizontal and vertical components of velocity and displacement to solve problems on projectiles
• apply calculus to the equations of motion to solve problems involving projectiles (ACMSM111, ACMSM113, ACMSM115)
• find the magnitude and direction of the velocity of a projectile at a given time or position, the range of a projectile on a horizontal plane and its greatest height reached