11–12Mathematics Extension 1 11–12 Syllabus

Define a vector as a quantity having both magnitude and direction

Associate vectors with directed line segments and recognise that a vector may have many directed line segments associated with it

Describe a position vector as a vector with its tail at the origin

Represent vectors graphically with and without graphing applications

Recognise and use the fact that two vectors are equal if they have the same magnitude and direction to solve problems

Use Cartesian coordinates to represent points in 2-dimensional (2D) and 3-dimensional (3D) space with and without graphing applications

Use the midpoint and distance formulas in two dimensions and three dimensions

Define unit vectors as vectors of magnitude 1

Define a scalar as a real number that is used to multiply a vector

Represent geometrically a scalar multiple of a vector in two dimensions and three dimensions with and without graphing applications

Perform multiplication of a vector by a scalar algebraically in component form

Perform addition and subtraction of vectors algebraically in component form, and verify, with and without graphing applications, that geometrically these are obtained using the triangle law or the parallelogram law

Describe the position of an object at a point in 2D space using a vector

Find the Cartesian equation of the path of an object, where the path is a straight line, parabola or circle

Express the change in an object’s position between two points as a displacement vector and recognise the magnitude of the displacement vector as the distance between the two points

Solve motion problems involving constant velocity using vectors

Solve relative velocity problems involving constant crosswind/cross-current using vector diagrams, and describe the direction of a vector where required

Find the position vector and the velocity vector of an object using integral calculus given its acceleration vector

Solve motion problems involving non-constant velocity using vectors

Recognise that the gravitational force on a mass may be regarded as a constant acting in a downwards direction when the motion of the object is restricted to a small region near the Earth’s surface

Model and analyse a projectile’s path where the projectile is a point and air resistance is negligible, subject to only acceleration due to gravity, assuming that the projectile is moving close to the Earth’s surface

Represent the motion of a projectile using vectors

Recognise that the horizontal and vertical components of the motion of a projectile can be represented by horizontal and vertical vectors

Derive and use the equations of motion of a projectile in vector form by splitting 2D motion into horizontal and vertical components to solve problems on projectiles

Find the Cartesian equation of the path of a projectile using parametric equations for the horizontal and vertical components of the displacement vector

Determine features of the flight of a projectile, including time of flight, maximum height, range, instantaneous velocity and impact velocity

Solve problems relating to the path of a projectile in which the initial velocity and/or angle of projection may be unknown, in a variety of contexts