Vectors

ME-V1 Introduction to Vectors

• define a vector as a quantity having both magnitude and direction, and examine examples of vectors, including displacement and velocity (ACMSM010)ACMSM010

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  • explain the distinction between a position vector and a displacement (relative) vector

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• define and use a variety of notations and representations for vectors in two dimensions (ACMSM014)ACMSM014

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  • use standard notations for vectors, for example: , $\overrightarrow{AB}$ and $a$

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  • represent vectors graphically in two dimensions as directed line segments

    not written yet · needs a worked example

  • define unit vectors as vectors of magnitude 1, and the standard two-dimensional perpendicular unit vectors and

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  • express and use vectors in two dimensions in a variety of forms, including component form, ordered pairs and column vector notation

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• perform addition and subtraction of vectors and multiplication of a vector by a scalar algebraically and geometrically, and interpret these operations in geometric terms AAM

  not written yet · needs a worked example

  • graphically represent a scalar multiple of a vector (ACMSM012)ACMSM012

    not written yet · needs a worked example

  • use the triangle law and the parallelogram law to find the sum and difference of two vectors

    not written yet · needs a worked example

  • define and use addition and subtraction of vectors in component form (ACMSM017)ACMSM017

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  • define and use multiplication by a scalar of a vector in component form (ACMSM018)ACMSM018

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• define, calculate and use the magnitude of a vector in two dimensions and use the notation for the magnitude of a vector

  not written yet · needs a worked example

  • prove that the magnitude of a vector, , can be found using:

    not written yet · needs a worked example

  • identify the magnitude of a displacement vector $\overrightarrow{AB}$ as being the distance between the points $A$ and $B$

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  • convert a non-zero vector into a unit vector by dividing by its length:

    not written yet · needs a worked example

• define and use the direction of a vector in two dimensions

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• define, calculate and use the scalar (dot) product of two vectors and AAM

  not written yet · needs a worked example

  • apply the scalar product, , to vectors expressed in component form, where

    not written yet · needs a worked example

  • use the expression for the scalar (dot) product, where $\theta$ is the angle between vectors and to solve problems

    not written yet · needs a worked example

  • demonstrate the equivalence, and use this relationship to solve problems

    not written yet · needs a worked example

  • establish and use the formula

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  • calculate the angle between two vectors using the scalar (dot) product of two vectors in two dimensions

    not written yet · needs a worked example

• examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021)ACMSM021

  not written yet · needs a worked example

• define and use the projection of one vector onto another (ACMSM022)ACMSM022

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• solve problems involving displacement, force and velocity involving vector concepts in two dimensions (ACMSM023) AAMACMSM023

  not written yet · needs a worked example

• prove geometric results and construct proofs involving vectors in two dimensions including to proving that: AAM

  not written yet · needs a worked example

  • the diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039)ACMSM039

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  • the midpoints of the sides of a quadrilateral join to form a parallelogram (ACMSM040)ACMSM040

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  • 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)ACMSM041

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• understand the concept of projectile motion, and model and analyse a projectile’s path assuming that:

  not written yet · needs a worked example

  • the projectile is a point

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  • the force due to air resistance is negligible

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  • the only force acting on the projectile is the constant force due to gravity, assuming that the projectile is moving close to the Earth’s surface

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• 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 AAM

  not written yet · needs a worked example

  • represent the motion of a projectile using vectors

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

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  • derive the horizontal and vertical equations of motion of a projectile

    not written yet · needs a worked example

  • understand and explain the limitations of this projectile model

    not written yet · needs a worked example

• use equations for horizontal and vertical components of velocity and displacement to solve problems on projectiles

  not written yet · needs a worked example

• apply calculus to the equations of motion to solve problems involving projectiles (ACMSM115) AAMACMSM115

  not written yet · needs a worked example