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KS4 Mathematics

Explore the basics of vectors in mathematics, including vector notation, arithmetic, and solving problems using vectors. Understand multiplying vectors by scalars, adding and subtracting vectors, and finding vector magnitudes.

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KS4 Mathematics

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  1. KS4 Mathematics S7 Vectors

  2. S7 Vectors Contents • A S7.2 Multiplying vectors by scalars • A S7.3 Adding and subtracting vectors • A S7.1 Vector notation S7.4 Vector arithmetic • A S7.5 Finding the magnitude of a vector • A S7.6 Using vectors to solve problems • A

  3. Vectors and scalars A vector is a quantity that has both size (or magnitude) and direction. Examples of vector quantities are: • displacement • velocity • force A scalar is a quantity that has size (or magnitude) only. Examples of scalar quantities are: • length • speed • mass

  4. Representing vectors A vector can be represented using a line segment with an arrow on it. For example, B A The 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 B We can write this vector as AB. A a When this is hand-written, the a is written as a This vector goes from the point A to the point B. Vectors can also be written using single letters in bold type. For example, we can call this vector a.

  6. Representing vectors 3 6 B AB A 6 = 3 To go from the point A to the point B we must move 6 units to the right and 3 units up. We can represent this movement using a column vector. This is the horizontal component. It tells us the number of units in the x-direction. This is the vertical component. It tells us the number of units in the y-direction.

  7. Representing vectors

  8. Equal vectors a b c d e f g Two vectors are equal if they have the same magnitude and direction. All of the following vectors are equal: They are the same length and parallel to each other.

  9. The negative of a vector B a Here is the vector A B A AB BA 5 = 2 –5 –a or –2 Suppose the arrow went in the opposite direction: How can we describe this vector? We can describe this vector as:

  10. The negative of a vector B a A B –a A x –x y a = –y If this is the vector a, this is the vector –a. The negative of a vector is the same length and has the same slope, but goes in the opposite direction. In general, –a = if then

  11. The negative of a vector

  12. The zero and unit vector 0 The zero vector is written as 0 or hand-written as The horizontal unit base vector, , is called i. The vertical unit base vector, , is called j. 1 0 0 1 A vector with a magnitude of 0 is called the zero vector. A vector with a magnitude of 1 is called a unit vector. The most important unit vectors are those that are horizontal and vertical. These are called unit base vectors.

  13. The unit base vectors j i For example, = 5i– 4j 5 –4 The unit base vectors, i and j, can be represented in a diagram as follows: Any column vector can easily be written in terms of i and j. The number of i’s tells us how many units are moved horizontally and the number of j’stell us how many units are moved vertically.

  14. The unit base vectors 7 4 = 4i+ j = 2 1 = –7j = –1 –5 = –i + 8j = 0 1 0 8 –3 –7 Write the following in terms of unit base vectors. Write the following in terms of column vectors. 7i+ 2j i– 3j –5i

  15. S7 Vectors Contents S7.1 Vector notation • A • A S7.3 Adding and subtracting vectors • A S7.2 Multiplying vectors by scalars S7.4 Vector arithmetic • A S7.5 Finding the magnitude of a vector • A S7.6 Using vectors to solve problems • A

  16. Multiplying vectors by scalars a 2a a = 3 6 2 4 2a = Remember, a scalar quantity has size but not direction. A scalar quantity can be represented by a single number. 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.

  17. Multiplying vectors by scalars In general, if the vector is multiplied by the scalar k, then –2 –6 3 × = 5 15 = k × kx x x y y ky For example, When a vector is multiplied by a scalar the resulting vector is either parallel to the original vector or lies on the same line. Can you explain why this is?

  18. Multiplying vectors by scalars

  19. Pairs – parallel vectors

  20. S7 Vectors Contents S7.1 Vector notation • A S7.2 Multiplying vectors by scalars • A • A S7.3 Adding and subtracting vectors S7.4 Vector arithmetic • A S7.5 Finding the magnitude of a vector • A S7.6 Using vectors to solve problems • A

  21. Adding vectors 3 5 8 Suppose a = b = and –2 3 1 b a a + b = a + b Adding two vectors is equivalent to applying one vector followed by the other. For example, Find a + b We can represent this addition in the following diagram:

  22. Adding vectors c a In general, if a = b = and d b a + c a + b = b + d When two or more vectors are added together the result is called the resultant vector. We can add two column vectors by adding the horizontal components together and adding the vertical components together.

  23. Adding vectors

  24. Subtracting vectors –2 4 Suppose a = and b = 3 4 –2 6 4 4 – –2 – = = 3 1 4 4 – 3 We can subtract two column vectors by subtracting the horizontal components and subtracting the vertical components. For example, Find a – b a – b =

  25. Subtracting vectors –2 4 a = and b = 3 4 b –b –b a a a – b 6 a – b = 1 To show this subtraction in a diagram, we can think of a – b as a + (–b).

  26. Adding and subtracting vectors

  27. The resultant vector

  28. The parallelogram law for vector addition 2 4 Suppose a = and b = 3 –1 D D a C C b b a + b a + b a + b AC BC AB AD DC AC = = + + = = b + a a + b b A A a a B B As you have seen, we can use a parallelogram to demonstrate the addition of two vectors. From this diagram we can see that D C A Also B Vector addition is commutative

  29. S7 Vectors Contents S7.1 Vector notation • A S7.2 Multiplying vectors by scalars • A S7.3 Adding and subtracting vectors • A S7.4 Vector arithmetic • A S7.5 Finding the magnitude of a vector • A S7.6 Using vectors to solve problems • A

  30. Vector arithmetic –1 –2 –1 5 5 a = b = and c = 4 4 6 3 3 5 + –1 3 × –2 4 a + c = + = = = 3 + 4 3 × 6 7 –2 –6 3b = 3 × = 6 18 –1 –2 –2 – –2 0 2c – b = 2 × – = = 4 6 8 – 6 2 Suppose, Find 1) a + c 2) 3b 3) 2c – b

  31. Vector equations Suppose, –4 6 a = and b = 7 –1 Find vector c such that 2c+ a = b Start by rearranging the equation to make c the subject. 2c + a = b 2c = b – a c = ½(b – a)

  32. Vector equations Suppose, –4 6 a = and b = 7 –1 6 – –4 10 ½ c = ½ = –1– 7 –8 5 c = –4 Find vector c such that 2c+ a = b Next, substitute a and b to find c, c = ½(b – a)

  33. A grid of congruent parallelograms

  34. Vectors on a tangram = b = a and I J H AE FC AF HJ HI CB HD IG F = = = = = = G b a E A tangram is an ancient Chinese puzzle in which a square ABCD is divided as follows: C D Suppose, Write the following in terms of a and b. 3b 2a– 4b –a b – a a– 2b 2b – 3a B A

  35. S7 Vectors Contents S7.1 Vector notation • A S7.2 Multiplying vectors by scalars • A S7.3 Adding and subtracting vectors • A S7.5 Finding the magnitude of a vector S7.4 Vector arithmetic • A • A S7.6 Using vectors to solve problems • A

  36. Finding the magnitude of a vector The magnitude of the vector is given by the length of the line. What is the magnitude of vector a? a We can find the magnitude using Pythagoras’ Theorem. The direction of the vector is given by the arrow on the line.

  37. Finding the magnitude of a vector 3 5 |a| = 32 + 52 = 34 The magnitude of the vector is given by the length of the line. What is the magnitude of vector a? a We often write |a| to represent the magnitude (or modulus) of a. = 5.83 (to 2 d.p.)

  38. Finding the magnitude of a vector

  39. Finding the magnitude of a vector 7 –5 2 a = b = 4 2 6 + 29 = 65 |b| = 52 + 22 |a| = 72 + 42 a + b = = 65 = = 40 = = 29 = |a + b| = 22 + 62 Suppose, and Find 1) |a| 8.06 (to 2 d.p.) 2) |b| 5.39 (to 2 d.p.) 3) |a|+|b| |a|+|b| = 13.45 (to 2 d.p.) 4) |a + b| 6.32 (to 2 d.p.)

  40. S7 Vectors Contents S7.1 Vector notation • A S7.2 Multiplying vectors by scalars • A S7.3 Adding and subtracting vectors • A S7.6 Using vectors to solve problems S7.4 Vector arithmetic • A S7.5 Finding the magnitude of a vector • A • A

  41. Using vectors to solve problems B C A 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: The line PQ is such that P is the mid-point of AB and Q is the mid-point of AC. P Use vectors to show that PQ is parallel to BC and that the length of BC is double the length of PQ. Q

  42. Using vectors to solve problems = 2 = = a AP AQ PQ PQ BC BC b Therefore, Let’s call vector a and vector b. –a + b B = b– a P –2a + 2b = 2b– 2a C A Q = 2(b– a) We can conclude from this that PQ is parallel to BC and that the length of BC is double the length of PQ.

  43. Using vectors to solve problems 6 N 8 N 8 N 6 N 6 N θ 8 N Suppose we have an object with two forces acting on it as shown: Find the magnitude and direction of the resultant force. The resultant force can be shown on the diagram as follows:

  44. Using vectors to solve problems The magnitude of the resultant force can be found using Pythagoras’ theorem. F 8 N 6 N 6 N θ 8 N If F is the resultant force then F2= 62 + 82 F2= 36 + 64 F2= 100 F = 10 N

  45. Using vectors to solve problems 6 tan θ= 8 The direction of the resultant force can be found using trigonometry. 10 N 8 N 6 N 6 N θ 8 N If θ is the angle that the resultant force makes with the horizontal then, θ = 36 .87° (to 2 d.p.) The resultant force has a magnitude of 10 N and makes an angle of 36.87° to the horizontal.

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