C4 maths: vectors

LESSON 1: DRAWING VECTORS

Starter: draw some examples of vectors on the whiteboard, and describe them.

Idea: in physics vectors are quantities with magnitude and direction, like displacement, velocity, acceleration, force, momentum. In maths, the idea of vector is more general: a vector of length n is a list of n numbers. We will learn how to do some calculations with them, and how to use vector calculations in geometry.

History: the theory of vectors was developed by physicists rather than pure mathematicians. Notably Josiah Willard Gibbs in the 1880s. He drew on the mathematical idea of quaternions.

Uses: vectors are used a lot in physics. But also, e.g. in economics. And in computer graphics.

Algorithm: multiplying vectors by scalars.

Activity: A vector a is defined by magnitude 2 and angle 45°. Draw a and 3a. What is a in (x,y) terms? What is 3a in (x,y) terms?

Activity on parallel vectors: Ex.5B Q.3,4

Algorithm: Adding vectors, numerically and geometrically.

Activity: Ex.5A Q.2,3. Draw the vectors a+b as well as working out |a+b|

Algorithm: to translate between a vector written as (r,θ) and a vector written as (x,y).

Notation: |a|. Algorithm: |a|=√(x2+y2)

Activity: Drawing vectors.

Summary: vectors are quantities which can be represented by an ordered list of numbers. Example: physical quantities with magnitude and direction. We can translate between a vector written as (r,θ) and a vector written as (x,y) and a vector drawn as an arrow. We can add vectors and multiply them by scalars


LESSON 2: USING VECTORS IN GEOMETRY. PARALLEL VECTORS. VECTORS IN TRIANGLES.

Starter: Draw and write some vectors which are parallel to each other.

Summary: draw vector addition and parallel vectors.

Activity: Ex.5B Q.1,2,4(a),4(b)

Homework: Ex.5B Q.5,6,7


LESSON 3: POSITION VECTORS. UNIT VECTORS. DESCRIBING VECTORS IN TERMS OF i AND j

Starter: If you have a triangle like on page 53, and the bottom right-hand corner is at the origin, what are the coordinates of the other two corners?

Algorithm: the position vector of a point equals the displacement from the origin to that point.

Activity: Ex.5C Q.1-3

Notation: a unit vector is a vector of unit magnitude

Algorithm: write vectors in xi+yj form.

Notation: column vectors

Activity: Ex.5D Q.1-2

Summary: Position vector=displacement from origin to the point. Unit vector=vector of unit magnitude. Write vectors as columns or xi+yj.


LESSON 4. 3-VECTORS

Starter: A student is sitting near the cabinet. A teacher is standing in front of the whiteboard. The student throws a ball of paper and hits the teacher on the forehead. What vector describes the displacement of the ball of paper?

Algorithm and notation: magnitude of 3-vector; i, j, k

Activity: Ex.5F Q.1-3

Homework: 1. Ex.5F Q.4-6. 2. use vectors to prove that the medians of a triangle (the lines connecting vertices with the midpoints of the opposite sides) meet at a point two-thirds down each median.

Follow the help given here.


LESSON 5: DOT PRODUCT

Starter: View dot product demonstration at Wolfram Demonstrations Project

Idea: Dot product measures how much two vectors work together. Example: work in mechanics. Dot product is not an ordinary multiplication, since the product is a scalar and not a vector, but it follows some multiplication rules.

Algorithm: Dot product = a1b1+…. = |a||b| cos θ

Idea of equivalence: consider case where b is along x-axis.

Proof of equivalence: see book p.70

Activity: Dot-product dating, shopping, and mechanics

Proof of cosine rule using vectors:

If ABC is a triangle, then:

.=

 ▇

Summary: Dot product measures how much two vectors work together. It’s a scalar. Two formulas for it.


LESSON 6: USING VECTORS TO DESCRIBE A LINE. VECTOR PERPENDICULAR TO TWO LINES.

Starter: Given a line across the classroom with direction vector b and starting point a, what equation describes all the points r on the line?

Algorithm: vector equations of line.

Activity: Ex.5H Q.1-3

Algorithm: Two lines across classroom. How do we find vector perpendicular to two lines? Example 28, p.73

Activity: Ex.5G Q.10

Summary: r=a+tb describes what line? What other vector equations describe the same line?

Homework: Ex.5G Q.1-3. Ex.5H Q.5


LESSON 7: DO TWO LINES INTERSECT? WHAT IS THE ANGLE BETWEEN THEM?

Starter: Given two lines across the classroom, how do we find whether they intersect?

Algorithm: three simultaneous equations, solve two, check the third

Activity: Ex.5I Q.1-2

Algorithm: angle of intersection from dot product of direction vectors

Note: acute and obtuse angle

Activity: Ex.5J Q.1-2

Summary: solve equations to find intersection; angle of intersection from dot product.

Homework: Ex.5K Q.1-5.