# Vectors

## Edexcel AS Further Mathematics Year 1: Further Pure 1: Vectors

#### Prerequisite Knowledge

The prerequisite knowledge from Edexcel A-Level Mathematics and Further Mathematics for learning about the vector product, scalar triple product, and applications of vectors in three-dimensional geometry typically includes:

1. Basic Algebra: Understanding algebraic manipulation, solving linear equations, and working with algebraic fractions.
2. Coordinate Geometry: Knowledge of the Cartesian coordinate system in two and three dimensions, including the concepts of points, lines, and planes.
3. Trigonometry: Familiarity with sine, cosine, and tangent functions, as well as the ability to use trigonometric identities.
4. Vectors in Two Dimensions: Prior understanding of vectors in two dimensions, including vector addition, subtraction, and scalar multiplication.
5. Dot Product of Vectors: Knowledge of how to calculate the dot product (scalar product) of two vectors and its geometric interpretation as the product of their magnitudes and the cosine of the angle between them.
6. Basic Calculus: While not always a direct prerequisite, having a foundational understanding of limits, derivatives, and integrals can help in understanding the geometric interpretations of vector calculus.

#### Success Criteria

• Understand and calculate the vector product (cross product) of two vectors, including the geometric interpretation of a×b as an area.
• Be able to use the scalar triple product a⋅(b×c) to calculate the volume of a parallelepiped and a tetrahedron.
• Apply vector methods to solve problems in three-dimensional geometry involving points, lines, and planes.
• Derive and use the equation of a line in the form (rab=0 to represent lines in space.
• Calculate direction ratios and direction cosines of a line to understand its orientation in three dimensions.

#### Teaching Points

• The concept and calculation of the vector product (cross product) for two vectors, emphasising its vector nature and perpendicularity to the plane of the original vectors.
• The geometric interpretation of the cross product as representing the area of a parallelogram spanned by the two vectors.
• How to compute the scalar triple product and understand its significance in determining the volume of a parallelepiped formed by three vectors.
• The application of the scalar triple product for calculating the volume of a tetrahedron using vectors.
• Introducing the equation of a line in three dimensions using the vector cross product, (rab=0, and explaining how it represents all points on the line.
• Teaching how to find direction ratios and direction cosines of a line and their use in determining the orientation of the line in three-dimensional space.

#### Common Misconceptions

• Cross Product as a Scalar: Misunderstanding the vector product as a scalar quantity rather than a vector perpendicular to the plane containing the original vectors.
• Direction of Cross Product: Confusing the direction of the cross product vector, forgetting to apply the right-hand rule or the left-hand rule depending on the convention used.
• Scalar Triple Product as a Vector: Believing that the scalar triple product results in a vector, not recognizing it as a scalar quantity representing volume.
• Independence of Order: Assuming that the cross product is commutative (i.e., a×b=b×a), not realizing that a×b=−(b×a).
• Equations of Lines and Planes: Mixing up the equations for lines and planes in three-dimensional space, especially the different forms of the equations.
• Direction Cosines and Ratios: Confusing direction cosines with direction ratios or not understanding their geometric and trigonometric significance.
• Cross Product for Parallel Vectors: Thinking that parallel vectors have a non-zero cross product, whereas it is actually zero.
• Magnitude of Areas and Volumes: Assuming any two vectors can form a parallelogram or that any three vectors can form a parallelepiped, regardless of their direction and whether they are coplanar or not.

## Vectors Resources

### Mr Mathematics Blog

#### Planes of Symmetry in 3D Shapes

Planes of Symmetry in 3D Shapes for Key Stage 3/GCSE students.

Use isometric paper for hands-on learning and enhanced understanding.

#### GCSE Trigonometry Skills & SOH CAH TOA Techniques

Master GCSE Math: Get key SOH-CAH-TOA tips, solve triangles accurately, and tackle area tasks. Ideal for students targeting grades 4-5.

#### Regions in the Complex Plane

Explore Regions in the Complex Plane with A-Level Further Maths: inequalities, Argand diagrams, and geometric interpretations.