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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

cos β = √ 41 41 It is conventional to label the direction cosines as and m so that 4 5 = √ . 5) and its position vector 4i + 5j shown in Figure 1. 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 = √ .3: Vectors . its direction ratio is a : b. y 5 P (4. m= √ a2 + b 2 a2 + b2 Engineering Mathematics: Open Learning Unit Level 1 2 9. m= √ 41 41 More generally we have the following result: Key Point For any vector r = ai + bj. Its direction cosines are a b = √ .5) r = 4i + 5j β α x O 4 Figure 1: −→ The direction ratio of the vector OP is deﬁned to be 4:5. −→ 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 (4.1. 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. Speciﬁcally.

Given a vector r = ai + bj + ck its direction ratios are a : b : c.3: Vectors . b) The direction ratio of AB is therefore 4:3. m and n and they are given by a b a = cos α = √ .Example Point A has coordinates (3. 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. P and Q have coordinates (−2. Answer 2. Direction Ratios and Cosines in Three Dimensions The concepts of direction ratio and direction cosines extend naturally to three dimensions. Its direction cosines are a b c = √ . 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. Consider Figure 2. −→ −→ a) Find the direction ratio of the vector P Q b) Find the direction cosines of P Q. 5). c) Find its direction cosines. 4) and (7. and m. 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 = cos β = √ . 8) respectively. −→ −→ a) Write down the vector AB. It is conventional to label direction cosines as . 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. The direction cosines are the cosines of the angles between the vector and each of the axes. d) Show that 2 + m2 = 1 Solution −→ −→ a) AB = b − a = 4i + 3j. m= √ . then 2 + m2 = 1 Now do this exercise 1. and point B has coordinates (7. b) Find the direction ratio of the vector AB. c) The direction cosines are 4 4 3 3 = √ = . 8).

3).c) γ α β y x Figure 2: More exercises for you to try 1. n) of AB. the direction cosines and the angles that the vector OP makes with each of the axes when P is the point with coordinates (2. −→ 2. A line is inclined at 60◦ to the x axis and 45◦ to the y axis. 3.3: Vectors . Find the direction ratios. Points A and B have position vectors a = −3i + 2j + 7k. m. e) Show that 2 + m2 + n2 = 1. and b = 3i + 4j − 5k respectively. Find its inclination to the z axis.4. Answer Engineering Mathematics: Open Learning Unit Level 1 4 9. Find −→ −→ −→ −→ a) AB b) |AB| c) the direction ratios of AB d) the direction cosines ( .b. z P(a.

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

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

2:4:3.0◦ . √ . a) 6i + 2j − 12k. 29 29 29 3. 60◦ or 120◦ .3: Vectors . √ 184 184 184 2 4 3 2. √ 6 2 −12 1. √ . √ . √ . 56. 68.1◦ . b) 184.2◦ . c) 6 : 2 : −12. Back to the theory 7 Engineering Mathematics: Open Learning Unit Level 1 9. d) √ . 42.

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