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and Direction Cosines ✒9.3 ✑

Introduction

Direction ratios provide a convenient way of specifying the direction of a line in three dimensional

space. Direction cosines are the cosines of the angles between a line and the coordinate axes.

In this Block we show how these quantities are calculated.

✬ ✩

① be familiar with two and three dimensional

Prerequisites vectors in cartesian form

**Before starting this Block you should . . . ② be familiar with the trigonometric ratios:
**

sine, cosine and tangent

✫ ✪

**Learning Outcomes Learning Style
**

After completing this Block you should be able To achieve what is expected of you . . .

to . . .

**✓ understand what is meant by the terms ☞ allocate suﬃcient study time
**

direction ratios and direction cosines of a

vector ☞ brieﬂy revise the prerequisite material

**✓ calculate these quantities given a vector ☞ attempt every guided exercise and most
**

in cartesian form of the other exercises

its direction ratio is a : b. 5) and its position vector 4i + 5j shown in Figure 1.1. m= √ a2 + b 2 a2 + b2 Engineering Mathematics: Open Learning Unit Level 1 2 9. Speciﬁcally. cos β = √ 41 41 It is conventional to label the direction cosines as and m so that 4 5 = √ . referring to Figure 1 these are cos α and cos β −→ √ √ Noting that the length of OP is 42 + 52 = 41 we can write 4 5 cos α = √ . We can interpret this as stating that to move in the direction of the line OP we must move 4 units in the x direction for every 5 units in the y direction. −→ The direction cosines of the vector OP are the cosines of the angles between the vector and each of the axes. The Direction Ratio and Direction Cosines Consider the point P (4.3: Vectors . m= √ 41 41 More generally we have the following result: Key Point For any vector r = ai + bj.5) r = 4i + 5j β α x O 4 Figure 1: −→ The direction ratio of the vector OP is deﬁned to be 4:5. Its direction cosines are a b = √ . y 5 P (4.

Its direction cosines are a b c = √ . n= √ a2 + b2 + c 2 a2 + b2 + c2 a2 + b2 + c 2 where 2 + m2 + n2 = 1 3 Engineering Mathematics: Open Learning Unit Level 1 9. b) The direction ratio of AB is therefore 4:3. Answer 2. 8). and m. d) Show that 2 + m2 = 1 Solution −→ −→ a) AB = b − a = 4i + 3j. Direction Ratios and Cosines in Three Dimensions The concepts of direction ratio and direction cosines extend naturally to three dimensions. 8) respectively. The direction cosines are the cosines of the angles between the vector and each of the axes. n = cos γ = √ a2 + b2 + c2 a2 + b 2 + c 2 a2 + b 2 + c 2 In general we have the following result: Key Point For any vector r = ai + bj + ck its direction ratios are a : b : c. This means that to move in the direction of the vector we must must move a units in the x direction and b units in the y direction for every c units in the z direction. m= √ . 5). −→ −→ a) Write down the vector AB. then 2 + m2 = 1 Now do this exercise 1. P and Q have coordinates (−2.Example Point A has coordinates (3. m = cos β = √ . Consider Figure 2. m= √ = 4 + 32 2 5 42 +32 5 d) 2 2 2 4 2 3 16 9 25 +m = + = + = =1 5 5 25 25 25 The ﬁnal result in the previous example is true in general: Key Point If and m are the direction cosines of a line lying in the xy plane.3: Vectors . m and n and they are given by a b a = cos α = √ . and point B has coordinates (7. −→ −→ a) Find the direction ratio of the vector P Q b) Find the direction cosines of P Q. Given a vector r = ai + bj + ck its direction ratios are a : b : c. It is conventional to label direction cosines as . 4) and (7. c) Find its direction cosines. c) The direction cosines are 4 4 3 3 = √ = . b) Find the direction ratio of the vector AB.

3: Vectors . Find −→ −→ −→ −→ a) AB b) |AB| c) the direction ratios of AB d) the direction cosines ( . and b = 3i + 4j − 5k respectively.3).c) γ α β y x Figure 2: More exercises for you to try 1. Find its inclination to the z axis. Answer Engineering Mathematics: Open Learning Unit Level 1 4 9.b. −→ 2. 3. A line is inclined at 60◦ to the x axis and 45◦ to the y axis. the direction cosines and the angles that the vector OP makes with each of the axes when P is the point with coordinates (2. n) of AB. e) Show that 2 + m2 + n2 = 1. Points A and B have position vectors a = −3i + 2j + 7k. Find the direction ratios. m.4. z P(a.

3 5 Engineering Mathematics: Open Learning Unit Level 1 9.3: Vectors . End of Block 9.

97 97 Back to the theory Engineering Mathematics: Open Learning Unit Level 1 6 9. 9 4 1. a) 9 : 4. √ . b) √ .3: Vectors .

0◦ . √ . a) 6i + 2j − 12k. 2:4:3.3: Vectors . √ 6 2 −12 1.2◦ . b) 184. √ 184 184 184 2 4 3 2. Back to the theory 7 Engineering Mathematics: Open Learning Unit Level 1 9. d) √ . 68. 42. √ . √ . 29 29 29 3. 60◦ or 120◦ .1◦ . 56. √ . c) 6 : 2 : −12.

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