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.
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
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.