General
- Maths S2 SB File Uploaded 24/01/22, 15:45
- S2 Maths TG File Uploaded 3/08/22, 14:43
UNIT7: VECTORS
Key unit competency
By the end of this unit, I will be able to solve the problems using operations on vectors.
Unit outline
• Concept of a vector, definition and properties
• Vectors in a cartesian plane
• Operations on vectors
• Magnitude of vectors as its length
7.1 Concept of a vector, definition and properties
Activity 7.1
1. A tourist arrived in Kigali and is to visit the national museum.
a) What two aspects of the journey must he know?
b) What is the name of the quantity that has these two aspects?
2. In pairs, roughly estimate the following :
a) The distance between your school and the nearest shopping centre.
b) The direction of your school from the nearest shopping centre.
c) How did you estimate the direction in (b) above?
3. The distance from Kigali to Butare is about 133.1 km. Uwase drove from Kigali to Butare and back.
a) What is the total distance covered by Uwase?
b) What is the total displacement?
c) What causes the correct values of part (a) and (b)?
A vector is any quantity that has both magnitude and direction. Two examples of vectors are force and velocity. Both force and velocity are in a particular direction. The magnitude of the force indicates the strength of the force. For velocity, the speed is the magnitude. Other examples include displacement, acceleration.
Note that magnitude and direction are the two properties of a vector. Quantities with magnitude only are called scalars. Examples of scalar quantities are; distance, mass, time.
Geometrically, we represent a vector as a directed line segment, whose length is proportional to the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head. This is shown in fig 7.1
In this book, we shall adopt bold small letter notation e.g. (a) for position vectors and bold capital letter notation e.g. (AB) for others.
For example, consider a triangle ABC in figure 7.2
Note
The opposite vector of vector a in Fig. 7.2 is -a and that of vector b is -b. A vector and its opposite vector eg a and -a have the equal magnitudes but point in opposite directions.
7.2 Vectors on Cartesian plane
7.2.1 Definition of a column vector
Activity 7.2
Given the points A (-3, 2), B (3, 5), C (0,-2), D (2, 2) and E (-3,-3). Plot them on a graph paper. Join the points appropriately to show the following vectors.
a) AB b) BC
c) CA d) ED
e) DA f) EB
Consider the vectors from point A to B in the figure 7.3
The vector AB and is a displacement of 6 units to the right and 4 units upwards.
Consider the vector from point A to C in the figure 7.4 below.
The number (x) at the top represents the horizontal displacement. The number y at the bottom represents the vertical displacement. When a displacement vector is written in this way it is called a column vector.
It is important to note that whenever the displacement is towards the right or upwards, it is a positive displacement while displacement to the left or downwards is negative
Example 7.1
a) Plot the points P (2, 3) and Q (7,4) and show vector PQ .
b) Write down the column vector PQ
Exercise 7.1
1. Draw cartesian planes and locate the vectors provided. Draw a line joining the two points and indicate the arrow showing the direction from the first to the last point.
a) A(2,2) and B(3,1)
b) P(5,3) and Q(2,2)
c) P(3,1) and Q(2,-3)
d) A(3,4) and B(4,10)
e) A(2,-3) and B(6,7)
f) A(5,1) and B(2,-3)
2. Without drawing, find the resultant vectors for the following points. a) A(2,0) and B(3,-11)
b) P(5,1) and Q(2,4)
c) P(-6,1) and Q(6,-3)
d) A(3,2) and B(4,-5)
7.2.3 Equivalent of vectors
Activity 7.3
Observe the figure 7.7 that is in a shape of a parallelogram and discuss the questions that follow
a) Compare the magnitudes and directions of vectors AB and DC. What do you notice? What is the name given to such vectors?
b) Compare the magnitudes and directions of vectors DA and BC. What do you notice
Two vectors that have equal magnitudes and same directions are called equal vectors.
Consider the pairs of vectors shown in the figure 7.8.
In figure 7.8 (a), vectors VU and WX have the same direction and magnitude hence VU=WX.
In figure 7.8(b) vectors VU and WX have same magnitudes but different directions. Because they are different in directions, the two vectors are not equal i.e. VU≠WX.
Vectors that are parallel and equal in magnitude but opposite in direction are called opposite vectors. Vectors VU and WX in Fig. 7.8 (b) are examples of opposite vectors.
(a) List all the vectors that are equivalent to:
(i) AC
(ii) GH
(b) Is vector AB equivalent to vector IJ? Give a reason.
7.2.4 Midpoints of a vector
Activity 7.4
1. Count the number of desks in your column and locate the middle desk. Identify the fellow students who sit there.
2. a) Using a scale of your choice on a graph paper, draw a cartesian plane and plot the points A(2,2) and B(6,6) and join them with a straight line.
b) On your cartesian plane, locate the point M which is mid-way between A and B by counting the number of squares.
c) How else could you have determined the coordinates of point M in (b)?
A point that bisects a vector equally is called midpoint. This point lies halfway on the vector.
Exercise 7.3
1. Calculate the coordinates of the midpoints of the line segment joining the following pairs of points.
(a) A (2, 1), B (5, 3)
(b) A (0, 3), B (2, 7)
(c) A (4, –1), B (4, 3)
(d) A (–2, 3), B ( 2, 1)
2. In each of the following cases,
(i) Find the column vector of PQ.
(ii) Hence or otherwise, find the coordinates of the midpoint.
(a) P (3, 0), Q (4, 3)
(b) P (–3, 1), Q (5, 1)
(c) P (–2, –1), Q (–12, –8)
(d) P (–9, 1) Q (12, 0)
(e) P (–8, 7), Q (–7, 8)
(f) P (–3, 2), Q (3, –2)
3. P is (1, 0), Q is (4, 2) and R is (5, 4).
(a) Use vector method to find the coordinates of S if PQRS is a parallelogram.
(b) Find the coordinates of the midpoints of the sides of the parallelogram.
7.3 Operations on vectors
7.3.1 Addition and subtraction of vectors by construction
Activity 7.5
a) i) In figure 7.11 (a), redraw the vector AB and BC. Let the head of vector AB meet the tail of vector BC at B.
ii) Draw a line joining A directly to C and indicate the direction of vector AC with double arrows.
iii) What is the representative vector for AC?
b) i) In Figure 7.11(b), redraw the vector AB and CB. Let the head of vector AB meet the head of vector CB at B.
ii) Draw a line joining A directly to C indicate the direction of vector AC with double arrows.
iii) What is the representative vector for AC? c) What do you notice in (a) (iii) and (b) (iii)?
In the figure 7.12 above, the end result of moving from A to B and then from B to C is the same as going from A to C directly. The end effect is to reach point C from point A.
Since the end result is same,
we write AC = AB + BC
c = a + b
The vector AC is called the resultant vector and is indicated by the double arrow. Similarly, if you go from A to B and then from B to C′, the effect is the same as going from A to C′ directly. The required effect is to reach point C′ from point A.
Since the end result is the same, we writeAC′=AB+BC′
c = a + –b
c = a – b
The vector AC′ is called the resultant vector and is indicated by the double arrow. It is also important to note that: AC′=AB – BC′
Let us consider another triangle ABC in figure 7.13. The triangle represents routes joining three towns A, B and C.
If you go from A to B, then from B to C, the effect is the same as going from A to C directly. The required effect is to reach town C from A.
Since the effect is the same,
then AB + BC = AC.
Vector AC is called the resultant vector of AB and BC. Such a vector is usually represented by a line segment with a double arrowhead.
Example 7.5
Using Fig. 7.14, write down the single vector equivalent to:
(a) AB + BC
(b) AE + ED
(c) BC + CD + DE
(d) ED + DC + CB
(e) AB + BA
(f) CD + DC
Use it to write down the single vector equivalent to
(a) ST + TU (b) TS – RS
(c) RS + ST (d) UR – SR
(e) UT – RT (f) UR + RT
(g) TS + ST
UR + RU
(i) RS + ST + TU (j) UT – ST + SR
(k) ST + TU + UR + RS (l) UT + TS + SR – RU
3. Draw a triangle STR and put arrows on its sides to show TS + SR = TR.
4. Draw a quadrilateral ABCD and on it show BC, CD and DA. State a single vector equivalent to BC + CD + DA.
5. A man walks 10 km in the NE direction, and then 4 km due north. Using an appropriate scale, draw a vector diagram showing the man’s displacement from his starting point. When he stops walking, how far from the starting point will he have walked?
6. Vectors a, b and c are such that a = b and b = c. What can you say about a and c?
7. Mr. Habimana’s family planned a sight seeing trip which was to take them from Kigali to Huye, then to Rubavu and back to Kigali. Draw a vector triangle to show their trip, using K to stands for Kigali, H for Huye and R for Rubavu. What vector does KH + HR + RK represent?
(a) Copy the figure. Mark with arrows and name two pairs of equal vectors.
(b) Write single vector to represent:
(i) PQ + QR
(ii) PS – RS
(iii) SP + PQ
(iv) SR + RQ
9. Use Fig. 7.18 to find the vector represented by the box to make the following vector equations true.
7.3.2 Addition and subtraction of column vectorsActivity 7.6Study the cartesian plane in Fig. 7.20 and answer the questions that follow.
a) State the column vectors for r, p and q.b) Find r + pc) State the mathematical relationship connecting r and p to qd) Find the column vectors for r – p.e) What does – r – p represent on the above Cartesian plane.
In general, when adding or subtracting vectors, the horizontal displacements and the vertical displacements are added or subtracted separately.
Use the graph in Fig. 7.22 shown next page to answer Questions 3 and 4 below.3. (a) Name all the vectors that are equal to AB and state their column vectors.(b) Name the vector which is equal to EF.(c) Is PQ equal to KL? Give a reason for your answer.(d) Is KL equal to QP? Why?(e) Simplify EF + FG + GH and give your answer as a column vector. (f) Name a resultant vector which is equal to NM.(g) Name three vectors which are equal to 2GL.4. Write all the vectors in Fig. 7.22 as column vectors.5. P is (5, 3), Q is (-4, 2) and R is (2, -3). Find the column vectors PQ, RQ and RP.
9. Simplify FG + GH giving your answer in column vector form. What is the name of the simplified vector?10. Draw diagrams on squared paper to show
7.4 Position vectorsOn a cartesian plane, the position of a point is given with reference to the origin, O, the intersection of the x- and y- axes. Thus, we can use vectors to describe the position of a point (Fig. 7.23).
From the origin, A is +2 units in the x direction and +4 units in the y direction. Thus, A has coordinates (2 , 4) and OA
Similarly, B is –3 units in the x direction and +1 units in the y direction. Thus, B has coordinates (–3 , 1) and O
C is –2 units in the x direction and –3 units in the y direction.
D is +3 units in the x direction and –6 units in the y direction.
OA, OB, OC and OD are known as position vectors of A, B, C, and D respectively. All position vectors have O as their initial point.Example 7.8Consider points A (2, 3) and B (9, 5) in the figure 7.24 below
Find the position vectors for A and B hence find AB
Example 7.10P has coordinates (2 , 3) and Q (7 , 5).(a) Find the position vector of (i) P (ii) Q(b) State the column vector for PO.(c) Find the column vector for PQ
Exercise 7.61. State the position vectors of the following points.(a) P(5, 3)(b) Q(2, 3)(c) R(-6, 8)(d) S(-3, -4)(e) T(0, 2)(f) U(-3, 0)2. State the coordinates of the points with the following position vectors.
3. Given A (6, 3) and B (-4, 9), find the coordinates of C when:(a) OC = OA + OB(b) OC + OB = OA4. Given that OR = OP + OQ, state the coordinates of R when the coordinates of P and Q are:(a) P (0, 1) and Q (3, 6)(b) P (-3, 2) and Q (5, 1) (c) P (-4, -3) and Q (2, 0)5. Use Fig. 7.26 to write down the position vectors of the marked points.
6. On squared paper, mark the points whose position vectors are given below.
(a) Write down the position vector of B and of C. (b) The position vector of point D
7.5 Multiplying vectors by a scalarActivity 7.7Kigali, Rwamagana and kayonza are along the same road. Assume the road is straight which runs east of Kigali. From Kigali to Rwamagana is 55 km and from Rwamagana to Kayonza is 35 km.a) Draw a sketch diagram showing the location of towns along the straight line.b) If the distance from Kigali to kayonza is a, express in terms of a, i) distance from Rwamagana to Kayonza. ii) distance from Kayonza to Rwamagana.
If vectors PQ = QR = RT, then PT = PQ + QR + RT = a + a + a = 3a. In the above case, the value 3 is considered as a scalar multiplied to a vector a. If we have a vector say a, it can be multiplied by a constant k to give the final result as the vector.
The scalar k can be any positive or negative number. Each component of the vector is multiplied by the scalar.
Note:When a vector is multiplied by a positive scalar, its direction does not change. However, when a vector is multiplied by a negative scalar, its direction is reversed.Exercise 7.7
7.6 The magnitude of a vectorActivity 7.8In groups, discuss the following:1. The distance from Kigali to Musanze is –100 km. Is this statement valid? Explain your answer.2. A man moved from a point K, 40 km due East to point M. At point M, he turned north and moved 30 km to point N. However, there is a direct route from K to N.a) Sketch the diagram showing the man’s movement.b) Find the shortest distance from K to N.c) If the distance from K to M is x units and the distance from M to N is y units. Express the distance K to N in terms of x and y.The distance between two points is a scalar quantity hence it has no direction. Its value can therefore never be negative. It is always positive.Activity 7.91. Discuss with your classmate how to find the magnitude between two points A(x1, y1) and B(x2, y2).2. Using a graph paper, draw points P(2,6) and Q(5,3). Find |PQ
We can also determine the magnitude of the vector between two points using the coordinates of the two points
In the figure 7.30 below, the magnitude of OA is the length of line OA and is denoted as |OA| or |a|
Since the magnitude of a vector is its length, the quantity is always a positive scalar.
From this example, multiplying a vector by a scalar, k, also multiplies its magnitude by k. In general, |kr| = k|r|.
Exercise 7.81. Let u = (1, 2) v = (3, 2) and w = (2,–1) be vectors in x–y plane. Find the following.(a) 3u + 2w(b) 2u + 3v(c) 2u – (w + v)(d) 3(u + 7w)(e) |3u – 2v|(f) |u-7v|(g) |4u – w|22. Let P (2, 4) and Q (3, 7) be two points on a straight line PQ. Find the midpoint of the line PQ. Find also the distance from P to Q.3. Compute the magnitudes of the following vectors in x-y plane.
7. Find the distance between points P and Q given that:a) P(1, -2) and Q(2, 1)b) P(2, -2) and Q(0, 4)c) P(0, -2) and Q(0, 4)d) P(3, -3) and Q(-4, 1)
9. Calculate the distances between the following pairs of points.(a) A (5, 0), B (10, 4)(b) C (7, 4), D (1, 12)(c) E (–1, –1), F(–5, –6)(d) P (4, –1), Q(–3, –4)(e) H (b, 4b), K(–2b, 8b)(f) M (–2m, 5m), N (–4m, –2m)10. State which of the following expressions represent the distance between the points A (a , b)
Unit summary1. A vector is any quantity that has both magnitude and direction. Examples of vector quantities are: displacement, velocity, acceleration and force.2. Properties of a vector quantity are magnitude and direction.3. When a displacement vector is written as AB = ( x, y ) x y is called a column vector.4. Vector can be denoted using different ways:i) Bold capital letter e.g. ABii) Capital letters with arrow e.g. ABiii) Position vector with bold and small letters e.g. a, b or a , b5. Null vector has no magnitude and direction. It is denoted as 0 or 0 . 6. Two vectors are equivalent if they have the same direction and equal magnitude.7. A point that bisects a vector equally is called a midpoint. It lies halfway on the vector.8. The vector sum of two or more vectors is called resultant vector. 9. All position vector have 0 as their initial position.10. When a vector a = (x,y) x y is multiplied by a scalar k, we obtain ka = k (x,y ) x y = kx ky11. When a vector is multiplied by a position scalar, its direction does not change. However, when a vector is multiplied by a negative scalar, its direction is reversed.
8. PQR is a straight line such that PQ = 2QR(a) Given P (6, 0) and R (4, 3), write down the column vectors for OP and OR.(b) If OP = p and OR = r, express RP, RQ and OQ in terms of p and r. Hence find the coordinates of point Q.9. Fig 7.32 shows triangle ABC. Use it to answer the questions that follow.
(a) What is the opposite vectors of:(i) AB(ii) BC(iii) AC(b) Given that A(1,3) and C(5,3). Find |AC|