Extension Vectors – 3D vectors

This lesson comprises four (4) master classes focusing on:

• Vectors in 3D
• Vector representation of lines

Content:

MEX-V1.1

• Understand and use a variety of notations and representations for vectors in three dimensions
  • define the standard unit vectors \( \underset{\sim}{i} \), \( \underset{\sim}{j} \) and \( \underset{\sim}{k} \)
  • express and use a vector in three dimensions in a variety of forms, including component form, ordered triples and column vector notation
• 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

MEX-V1.2

• Define, calculate and use the magnitude of a vector in three dimensions
  • establish that the magnitude of a vector in three dimensions can be found using: \( |x \underset{\sim}{i}+y \underset{\sim}{j}+z \underset{\sim}{k}|=\sqrt{x^2+y^2+z^2} \)
  • 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 scalar (dot) product of two vectors in three dimensions
  • define and apply the scalar product \( \underset{\sim}{u} \cdot \underset{\sim}{v} \) to vectors expressed in component form, where \( \underset{\sim}{u} \cdot \underset{\sim}{v}=x_1x_2+y_1y_2+z_1z_2 \) ,  \( \underset{\sim}{u}=x_1 \underset{\sim}{i}+y_1 \underset{\sim}{j}+z_1 \underset{\sim}{k} \) and \( \underset{\sim}{v}=x_2 \underset{\sim}{i}+y_2 \underset{\sim}{j}+z_2 \underset{\sim}{k} \)
  • extend the formula \( \underset{\sim}{u} \cdot \underset{\sim}{v}=| \underset{\sim}{u} || \underset{\sim}{v}| \cos \theta \) for three dimensions and use it to solve problems
• Prove geometric results in the plane and construct proofs in three dimensions

MEX-V1.3

• Use Cartesian coordinates in two and three-dimensional space
• Recognise and find the equations of spheres
• Use vector equations of curves in two or three dimensions involving a parameter, and determine a corresponding Cartesian equation in the two-dimensional case, where possible
• Understand and use the vector equation \( \underset{\sim}{r}=\underset{\sim}{a}+\lambda \underset{\sim}{b} \) of a straight line through points \( A \) and \( B \) where \( R \) is a point on \( AB \), \( \underset{\sim}{a}=\vec{OA} \), \( \underset{\sim}{b}=\vec{AB} \), \( \lambda \) is a parameter and \( \underset{\sim}{r}=\vec{OR} \)
• 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
• Determine when two lines in vector form are parallel
• Determine when intersecting lines are perpendicular in a plane or three dimensions
• Determine when a given point lies on a given line in vector form