Vectors Lesson 1 Aims: • To understand what a vector is. • To be able to add and subtract vectors and multiply them by a scalar. • To know the unit vectors i, j, and k. • To know what a position vector is.

Representing vectors A vector is a quantity that has both size (or m______________) and direction. We can write this vector as . Or hand-written as a Or by using a single letters in bold type. E.g. a A B We can represent this movement using a c________ vector. 3 6 This example shows the vector from point A to point B. Ask pupils to suggest ways to describe this vector. For example, we could express it in terms of a movement through a given number of units up and to the right. We could also describe it in term of the length of the line and the angle it is moved through, as we do with bearings. Stress that when a is hand-written we must put a squiggle underneath to show that it is a vector. We can also represent vectors in 3D: C 5 This number of units moved in the z-direction. –3 z –2 y D x

Adding vectors Adding two vectors is equivalent to applying one vector followed by the other. For example, Suppose a = 5 3 and b –2 Find a + b We can represent this addition in the following diagram: Explain that adding these two vectors is like moving right 5 and up 3 and then moving right 3 and down 2. The net effect is a movement right 8 and up 1. Point out that we can add the horizontal components together to get the horizontal component of the resultant vector (5 + 3 = 8) and we can add the vertical components together to get the vertical component of the resultant vector (3 + –2 = 1). In the vector diagram the start of vector b is placed at the end of vector a. The resultant vector, a + b, goes from the start of a to the end of b. b a a + b = a + b

Subtracting vectors We can think of the subtraction of two vectors, a – b, as a + (–b). For example, suppose and . a b –b a –b a – b Explain that to draw a diagram of a – b we draw vector a followed by vector –b. The resultant vector a – b goes from the beginning of a to the end of –b. Establish again that subtracting the horizontal components gives 4 – –2 = 6 and subtracting the vertical components gives 4 – 3 = 1. a – b = Note: Two vectors are equal if they have the same magnitude and direction.

Multiplying vectors by scalars Remember, a scalar quantity can be represented by a single number. It has size but not direction. A vector can be multiplied by a scalar. For example, suppose the vector a is represented as follows: The vector 2a has the same direction but is twice as long. 2a a Point out that when the line is twice as long the horizontal and vertical components are doubled. 2a = So we can say that these two vectors are p________________.

A grid of congruent parallelograms In this activity the vectors a and b form the basis of the grid. Explain that this means that any point on the grid can be expressed in terms of these two vectors. This is true for any two non-parallel vectors. On mini whiteboards

The unit base vectors k j i In three dimensions, unit base vectors, i, j, and k, that runs parallel to the x, y and z-axis. They have a magnitude (length) of 1 unit. z-axis i is , j is and k is . y-axis k j i x-axis For example, the three-dimensional vector can be written in terms of i, j and k as Point out that the z-axis is usually drawn pointing vertically upwards with the x-and y-axes drawn in the horizontal plane. Vectors written in terms of the unit base vectors i, j and k are usually said to be written in c_____________________ form.

Vector arithmetic Given that a = 2i – 4j + k and b = j + 2k find 3a – 2b. 3a – 2b = Which of these is parallel to ?

Position vectors A p______________ vector is a vector that is fixed relative to a fixed origin O. For example, O P The position vector of the point P is given by p The position vector of the point Q is given by Stress that although there are infinitely many vector that have the same direction and magnitude as p and q, these vectors are unique in that they are fixed at the origin. q Q

The mid-point of a line Let M be the mid-point of the line PQ. P What is the position vector of the point M? M Warn students that the position vector of the mid-point of PQ is not 1/2(q – p).

On w/b Do exercise 9A page 101 then match cards. Let M be the 2 thirds along the line QP. O P p q Q What is the position vector of the point M? M Warn students that the position vector of the mid-point of PQ is not 1/2(q – p).

A: (1,-2,3) B: (-1,-3,2) OC: i+2j OD: i+2k

A: (1,-2,3) B: (-1,-3,2) OC: i+2j OD: i+2k