3 Vectors and scalarsA vector is a quantity that has both size (or magnitude) and direction.Examples of vector quantities include:displacementvelocityforceA scalar is a quantity that has size (or magnitude) only.Examples of scalar quantities include:lengthspeedmass
4 Representing vectorsA vector can be represented using a line segment with an arrow on it.For example, the vector that goes from the point A to the point B can be represented by the following directed line segment.BAThe magnitude of the vector is given by the length of the line.The direction of the vector is given by the arrow on the line.
5 Representing vectors We can write this vector as . Vectors can also be written using single letters in bold type.For example, we can call this vector a.When this is hand-written, the a is written asaTo go from the point A to the point B we must move 6 units to the right and 3 units up.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.AB36
6 Representing vectorsWe can represent this movement using a column vector.This component tells us the number of units moved in the x-direction.This component tells us the number of units moved in the y-direction.We can also represent vectors in three dimensions relative to a three dimensional coordinate grid:A third axis, the z-axis, is added at right angles to the xy-plane.Conventionally, we show the z-axis pointing vertically upwards with the xy-plane horizontal.
7 Representing vectorsFor example, consider the following three-dimensional vectorTo go from the point C to the point D we must moveC5–35 units in the x-direction,z–2y–3 units in the y-directionD–2 units in the z-direction.xThis three-dimensional vector can be written in column vector form as:This component tells us the number of units moved in the x-direction.This component tells us the number of units moved in the y-direction.This component tells us the number of units moved in the z-direction.
8 Equal vectorsTwo vectors are equal if they have the same magnitude and direction.For example, in the following diagram:ABCDandGeneral displacement vectors that are not fixed to any point are often called free vectors.
9 The negative of a vector BHere is the vectorABSuppose the arrow went in the opposite direction, from B to A:Recall that inverse translations map objects back to there starting points. Therefore, applying a vector followed by its inverse results in the zero vector, 0.This is the negative (or inverse) of the vectorWe can describe this new vector as:–aor
10 The negative of a vector In general,IfthenAnd in three-dimensions,IfthenEmphasize that the components of the negative of a given column vector has the same numbers with different signs.
11 The zero and unit vector A vector with a magnitude of 0 is called the zero vector.The zero vector is written as 0 or hand-written asA vector with a magnitude of 1 is called a unit vector.The most important unit vectors are those that run parallel to the x- and y-axes. These are called unit base vectors.The horizontal unit base vector, , is called i.The unit base vector i is the vector that is one unit in the x direction. The unit base vector j is the vector that is one unit in the y direction.The vertical unit base vector, , is called j.
12 The unit base vectorsThe unit base vectors, i and j, run parallel to the x- and y-axes.y-axisjx-axisiAny column vector can easily be written in terms of i and j.For example,The number of i’s tells us how many units are moved horizontally, and the number of j’s tells us how many units are moved vertically.
13 The unit base vectorsIn three dimensions, we introduce a third unit base vector, k, that runs parallel to the z-axis.z-axisi is , j is and k isy-axiskjiFor example, the three-dimensional vector can be written in terms of i, j and k asx-axisPoint out that the z-axis is usually drawn pointing vertically upwards with the x-and y-axes drawn in the horizontal plane.–i + 6j –3kVectors written in terms of the unit base vectors i, j and k are usually said to be written in component form.
15 Finding the magnitude of a vector The magnitude (or modulus) of a vector is given by the length of the line segment representing it.For example, suppose we have the vectoraABThe magnitude of this vector is written as or a .We can calculate this using Pythagoras’s Theorem.Establish that the magnitude of a vector is not affected by its direction. Magnitude, like length, is always positive. Point out that also that when finding the length of a line we can ignore negative directions because any number squared is a positive number.
17 Finding the magnitude of a vector The magnitude of a three-dimensional vector can be found by applying Pythagoras’s Theorem in three dimensions.For example, suppose we have the vectorThe magnitude of this vector is given by
19 The distance between two points If we are given the coordinates of two points and we are asked to find the distance between them we use Pythagoras’ Theorem in the same way. For example,Find the distance between the points with coordinates P(–4, 7, –2) and Q(5, 9, –8).If d is the distance between the points then, using Pythagoras’ Theorem in three dimensions gives:d2 = (–4 – 5)2 + (7 – 9)2 + (–2 – –8)2d2 =d2 = 121 d = 11In general, if d is the distance between the points (x1, y1, z1) and (x2, y2, z2) thend2 = (x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2
20 Unit vectorsRemember, if the magnitude of a vector is 1 it is called a unit vector.It is possible to find a unit vector parallel to any given vector, a, by dividing the vector by its magnitude.The unit vector parallel to the vector a is denoted bySo, in general,Find a unit vector parallel to b = 4i – j + k
22 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.2aPoint out that when the line is twice as long the horizontal and vertical components are doubled.a
23 Multiplying vectors by scalars In general, if the vector is multiplied by the scalar k, then:For example,Discuss the fact that when a vector is multiplied by a positive scalar it changes its length but not its direction. Vectors with the same direction have the same slope or gradient and are therefore parallel. They could also lie on the same line.Ask pupils what would happen if a vector was multiplied by a negative scalar. Establish that the resulting vector would still be parallel to, or on the same line as, the original vector but that the arrow would point in the opposite direction.When a vector is multiplied by a scalar the resulting vector lies either parallel to the original vector or on the same line.
24 Multiplying vectors by scalars Modify vector a by dragging on its end-points. Observe how the scalar multiples of the vector change.
26 Adding vectorsAdding two vectors is equivalent to applying one vector followed by the other.For example, suppose andWe can represent the addition of these two vectors in the following diagram:bExplain 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.aa + b
27 Adding vectorsStress that going from A to B to C is the same as going from A to C directly. This is often called the triangle law.The horizontal and vertical components of each vector can be hidden if required.
28 Adding vectorsWhen two or more vectors are added together the result is called the resultant vector.In general, if and , thenGiven that a = 2i + 6j – k and b = –i + 2j + 7k, find a + b.a + b = (2 –1)i + (6 + 2)j + (–1 + 7)k= i + 8j + 6k
29 Subtracting vectorsWe can think of the subtraction of two vectors, a – b,as a + (–b).For example, suppose andab–ba–bExplain 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
30 Adding and subtracting vectors Drag the points A, B, C and D to demonstrate the addition and subtraction of vectors.
31 The parallelogram law for adding vectors Start by clicking on vectors BD and CD to shade them out. The vectors a and b are then originating from the point A.Explain that when two vectors start from the same point, the sum of the two vectors, the resultant vector, can be found by completing a parallelogram. The resultant vector goes from the starting point to the opposite corner of the parallelogram.Give the real-life example of two forces represented by vectors a and b acting on a body at A. The resultant of these two forces is represented by a + b as shown from A.The vectors in the diagram can be faded or revealed by clicking on them. Use this to show that a + b is equivalent to b + a.This is sometimes called the parallelogram law.Ask pupils to explain why the other diagonal in the parallelogram DB = a – b.Arranging the vectors in a parallelogram shows that vector addition is commutative, in other words a + b = b + a.Going from A to C via D is equivalent to going from A to C via B.
32 Vector arithmeticWe have seen that vectors can be multiplied by scalars, added and subtracted. We have also seen that vector addition is commutative.We can use this to add and subtract any given multiple of a vector given in component or column vector form. For example,Given that a = 2i – 4j + k and b = j + 2k find 3a – 2b.3a – 2b = 3(2i – 4j + k ) – 2(j + 2k )= 6i – 12j + 3k – 2j – 4k= 6i – 14j – k
33 Vector arithmetic Suppose that and . Find vector c such that 2c + a = b.Suppose that andStart by rearranging the equation to make c the subject.2c + a = b2c = b – ac = (b – a)Explain that since vectors follow many of the rules of arithmetic we can apply these rules to solve equations involving vectors. Stress, however, that we cannot multiply or divide a vector by another vector.
34 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.
35 Using vectors to solve problems We can use vectors to solve many problems involving physical quantities such as force and velocity.We can also use vectors to prove geometric results.For example, suppose we have a triangle ABC as follows:ABCThe line PQ is such that P is the mid-point of AB and Q is the mid-point of AC.PUse vectors to show that PQ is parallel to BC and that the length of BC is double the length of PQ.Q
36 Using vectors to solve problems Let’s call vector a and vector b.BPaAbQCExplain that to go from B to C we can go from B to A and then from A to C. This is equivalent to –2a + 2b.Remind pupils that when one vector is a scalar multiple of another we can conclude that the two vectors are parallel.Therefore,We can conclude from this that PQ is parallel to BC and that the length of BC is double the length of PQ.
37 Collinear pointsThree or more points are said to be collinear if they lie on the same line. For exampleProve that the three points A(–3, 2, 6), B(1, 4, –2) and C(1, 5, –6) are collinear.Since AB is a scalar multiple of BC the two lines must be parallel. They also have the point B in common and so the points A, B and C must be collinear.
39 Position vectorsA position vector is a vector that is fixed relative to a fixed origin O.For example, suppose the point P has coordinates (4, 6).OPThe position vector of the point P is given bypNow, suppose the point Q has coordinates (3, –2).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.The position vector of the point Q is given byqQWrite the vector PQ as a column vector.
40 Position vectorsTo get from the point P to the point Q, we have to go from P to O …… and then from O to Q.P–pOPpqQSoq – pOqQWe can check this using the vector diagram.
41 Position vectorsIn general, if A is the point with coordinates (x1, y1) and B is the point with coordinate (x2, y2) we can write the position vectorsandThe vector is given byPoint out that this generalization can also be applied to points given in three dimensions.The vector can also be written in terms of i and j as
42 The mid-point of a lineReturning to our example using the points P(4, 6) and Q(3, –2):OPpqQLet M be the mid-point of the line PQ.What is the position vector of the point M?MWarn students that the position vector of the mid-point of PQ is not 1/2(q – p).
43 The mid-point of a line P and so p M m O q Q Warn students that the position vector of the mid-point of PQ is not 1/2(q – p).In general, if points A and B have position vectors a and b, then the position vector of the mid-point of the line AB is given by: