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Direction Ratios  ✏

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) r = 4i + 5j β α x O 4 Figure 1: −→ The direction ratio of the vector OP is defined to be 4:5.3: Vectors . cos β = √ 41 41 It is conventional to label the direction cosines as and m so that 4 5 = √ . The Direction Ratio and Direction Cosines Consider the point P (4. Its direction cosines are a b = √ . m= √ 41 41 More generally we have the following result: Key Point For any vector r = ai + bj.1. 5) and its position vector 4i + 5j shown in Figure 1. Specifically. −→ The direction cosines of the vector OP are the cosines of the angles between the vector and each of the axes. 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 = √ . y 5 P (4. 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. m= √ a2 + b 2 a2 + b2 Engineering Mathematics: Open Learning Unit Level 1 2 9.

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

z P(a.3: Vectors . 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. Find the direction ratios. Answer Engineering Mathematics: Open Learning Unit Level 1 4 9. e) Show that 2 + m2 + n2 = 1.3).c) γ α β y x Figure 2: More exercises for you to try 1. 3. and b = 3i + 4j − 5k respectively.b. −→ 2. Find −→ −→ −→ −→ a) AB b) |AB| c) the direction ratios of AB d) the direction cosines ( . m.4. Find its inclination to the z axis. A line is inclined at 60◦ to the x axis and 45◦ to the y axis. Points A and B have position vectors a = −3i + 2j + 7k.

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

b) √ . a) 9 : 4. √ . 97 97 Back to the theory Engineering Mathematics: Open Learning Unit Level 1 6 9.3: Vectors . 9 4 1.

29 29 29 3. √ 184 184 184 2 4 3 2.0◦ . √ . √ . Back to the theory 7 Engineering Mathematics: Open Learning Unit Level 1 9. 2:4:3. c) 6 : 2 : −12. b) 184. 56.2◦ .3: Vectors . √ 6 2 −12 1. 68. 60◦ or 120◦ . √ . 42. d) √ .1◦ . √ . a) 6i + 2j − 12k.