Extension Vectors – Introduction

This lesson comprises two (2) master classes focusing on:

• Vectors
• Operations with vectors
• Projectile motion

Content:

ME-V1.1

• Define a vector as a quantity having both magnitude and direction, and examine examples of vectors, including displacement and velocity
  • explain the distinction between a position vector and a displacement (relative) vector
• Define and use a variety of notations and representations for vectors in two dimensions
  • use standard notations for vectors, for example: \( \underset{\sim}{a} \), \( \vec{AB} \) and \( \mathbf{a} \)
  • represent vectors graphically in two dimensions as directed line segments
  • define unit vectors as vectors of magnitude 1, and the standard two-dimensional perpendicular unit vectors \( \underset{\sim}{i} \) and \( \underset{\sim}{j} \)
  • express and use vectors in two dimensions in a variety of forms, including component form, ordered pairs and column vector notation
• 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
  • use the triangle law and the parallelogram law to find the sum and difference of two vectors
  • define and use addition and subtraction of vectors in component form
  • define and use multiplication by a scalar of a vector in component form

ME-V1.2

• Define, calculate and use the magnitude of a vector in two dimensions and use the notation \( | \underset{\sim}{u} | \) for the magnitude of a vector \( \underset{\sim}{u}=x \underset{\sim}{i}+y \underset{\sim}{j} \)
  • prove that the magnitude of a vector, \( \underset{\sim}{u}=x \underset{\sim}{i}+y \underset{\sim}{j} \), can be found using: \( |\underset{\sim}{u}|=|x \underset{\sim}{i}+y \underset{\sim}{j}|=\sqrt{x^2+y^2} \)
  • identify the magnitude of a displacement vector \( \vec{AB} \) as being the distance between the points \( A \) and \( B \)
  • convert a non-zero vector \( \underset{\sim}{u} \) into a unit vector \( \hat{\underset{\sim}{u}} \) by dividing by its length: \( \hat{\underset{\sim}{u}}=\frac{\underset{\sim}{u}}{|\underset{\sim}{u}|} \)
• Define and use the direction of a vector in two dimensions
• Define, calculate and use the scalar (dot) product of two vectors \( \underset{\sim}{u}=x_1 \underset{\sim}{i}+y_1 \underset{\sim}{j} \) and \( \underset{\sim}{v}=x_2 \underset{\sim}{i}+y_2 \underset{\sim}{j} \)
  • apply the scalar product, \(\underset{\sim}{u} \cdot \underset{\sim}{v} \), to vectors expressed in component form, where \( \underset{\sim}{u} \cdot \underset{\sim}{v}=x_1x_2+y_1y_2 \)
  • use the expression for the scalar (dot) product, \( \underset{\sim}{u} \cdot \underset{\sim}{v}=|\underset{\sim}{u}∣∣\underset{\sim}{v}| \cos \theta \) where \( \theta \) is the angle between vectors \( \underset{\sim}{u} \) and \( \underset{\sim}{v} \) to solve problems
  • demonstrate the equivalence, \( \underset{\sim}{u} \cdot |\underset{\sim}{u}||\underset{\sim}{v}| \cos \theta =x_1x_2+y_1y_2 \) and use this relationship to solve problems
  • establish and use the formula \( \underset{\sim}{v} \cdot \underset{\sim}{v}=|\underset{\sim}{v}|^2 \)
  • calculate the angle between two vectors using the scalar (dot) product of two vectors in two dimensions
• Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular
• Define and use the projection of one vector onto another
• Solve problems involving displacement, force and velocity involving vector concepts in two dimensions
• Prove geometric results and construct proofs involving vectors in two dimensions including proving that:
  • the diagonals of a parallelogram meet at right angles if and only if it is a rhombus
  • the midpoints of the sides of a quadrilateral join to form a parallelogram
  • 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

ME-V1.3

• Understand the concept of projectile motion, and model and analyse a projectile’s path assuming that:
  • the projectile is a point
  • the force due to air resistance is negligible
  • 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
• 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 using vectors
  • recognise that the horizontal and vertical components of the motion of a projectile can be represented by horizontal and vertical vectors
  • derive the horizontal and vertical equations of motion of a projectile
  • understand and explain the limitations of this projectile model
• 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