Lesson Video: Direction Angles and Direction Cosines Mathematics • Third Year of Secondary School

Video Transcript

In this video, we will learn how to find direction angles and direction cosines for a given vector in space. We will begin by considering the three-dimensional coordinate grid. We know that in three-dimensional space, we have the 𝑥-, 𝑦-, and 𝑧- or 𝑧-axis. These are all perpendicular or at right angles to one another.

Let’s assume we have the vector 𝐯 in the direction shown, where vector 𝐯 has components 𝐯 sub 𝑥, 𝐯 sub 𝑦, and 𝐯 sub 𝑧. It is also important to recall that the unit vectors 𝐢, 𝐣, and 𝐤 act in the 𝑥-, 𝑦-, and 𝑧-direction, respectively. The first direction angle we need to be aware of is 𝛼. This is the angle between the unit vector 𝐢 and 𝐯. Our second angle is the angle 𝛽, which is the angle between the unit vector 𝐣 and vector 𝐯. Finally, we have the angle 𝛾, which is the angle between the unit vector 𝐤 and vector 𝐯. These are the three direction angles 𝛼, 𝛽, and 𝛾.

Now let’s consider the direction cosines. We know that in right-angle trigonometry, the cos of any angle is equal to the adjacent over the hypotenuse. The direction cosines are simply the cos of our three angles, cos of 𝛼, cos of 𝛽, and cos of 𝛾. cos of 𝛼 is equal to the 𝑥-component of the vector, in this case, 𝐯 sub 𝑥, divided by the magnitude of vector 𝐯. In the same way, the cos of angle 𝛽 is equal to 𝐯 sub 𝑦 divided by the magnitude of 𝐯. And finally, the cos of angle 𝛾 is equal to 𝐯 sub 𝑧 divided by the magnitude of 𝐯. The cos of 𝛼, cos of 𝛽, and cos of 𝛾 are known as the direction cosines.

By taking the inverse cos of both sides of these three equations, we get expressions for angle 𝛼, 𝛽, and 𝛾. 𝛼 is equal to the inverse cos of 𝐯 sub 𝑥 over the magnitude of 𝐯. Likewise, angle 𝛽 is equal to the inverse cos of 𝐯 sub 𝑦 divided by the magnitude of 𝐯. The angle 𝛾 is equal to the inverse cos of 𝐯 sub 𝑧 over the magnitude of 𝐯. We recall that if vector 𝐯 has components 𝐯 sub 𝑥, 𝐯 sub 𝑦, and 𝐯 sub 𝑧, then the magnitude of vector 𝐯 is equal to the square root of 𝐯 sub 𝑥 squared plus 𝐯 sub 𝑦 squared plus 𝐯 sub 𝑧 squared. Note that this magnitude of a vector is also sometimes called the norm. In our first question, we will find a vector given its direction angles.

Find vector 𝐀 whose norm is 41 and whose direction angles are 135 degrees, 120 degrees, and 60 degrees.

We know that the norm of a vector is its magnitude and that the direction vectors 𝛼, 𝛽, and 𝛾 are the angles between the unit vectors 𝐢, 𝐣, and 𝐤 and the vector 𝐯. We also know that the direction cosines are such that cos 𝛼 is equal to 𝐯 sub 𝑥 over the magnitude of 𝐯, cos 𝛽 is equal to 𝐯 sub 𝑦 over the magnitude of 𝐯, and cos 𝛾 is equal to 𝐯 sub 𝑧 over the magnitude of 𝐯, where the vector 𝐯 has components 𝐯 sub 𝑥, 𝐯 sub 𝑦, and 𝐯 sub 𝑧. Substituting in our values for 𝛼 and the magnitude, we see that the cos of 135 degrees is equal to 𝐀 sub 𝑥 over 41. We can then multiply both sides of this equation by 41. 𝐀 sub 𝑥 is equal to negative 41 root two over two. This is the 𝑥-component of our vector 𝐀.

In the same way, considering angle 𝛽, we have the cos of 120 degrees is equal to 𝐀 sub 𝑦 over 41. Once again, we multiply both sides of the equation by 41 such that 𝐀 sub 𝑦 is equal to negative 41 over two. Finally, we have the cos of 60 degrees, angle 𝛾, is equal to 𝐀 sub 𝑧 divided by 41. 𝐀 sub 𝑧 is therefore equal to 41 multiplied by the cos of 60 degrees, which in turn is equal to 41 over two. This means that vector 𝐀 has components negative 41 root two over two, negative 41 over two, and 41 over two. This is the vector whose magnitude or norm is 41 and direction angles are 135, 120, 60 degrees.

Before looking at our next example, we will consider a formula that links the direction cosines. cos squared 𝛼 plus cos squared 𝛽 plus cos squared 𝛾 is equal to one. From our knowledge of the direction cosines, let’s consider why this is true. We know that the cos of 𝛼, 𝛽, and 𝛾 are as shown. Squaring both sides of this equation, we see that cos squared 𝛼 is equal to 𝐯 sub 𝑥 squared divided by the magnitude of 𝐯 squared. Doing the same to our other two equations, we get cos squared 𝛽 is equal to 𝐯 sub 𝑦 squared over the magnitude of 𝐯 squared and cos squared 𝛾 is equal to 𝐯 sub 𝑧 squared over the magnitude of 𝐯 squared.

We can now add the three expressions on the right-hand side as this would be equal to cos squared 𝛼 plus cos squared 𝛽 plus cos squared 𝛾. As all three fractions have the same denominator, this is equal to 𝐯 sub 𝑥 squared plus 𝐯 sub 𝑦 squared plus 𝐯 sub 𝑧 squared all divided by the magnitude of 𝐯 squared. We know that the magnitude of 𝐯 is equal to the square root of 𝐯 sub 𝑥 squared plus 𝐯 sub 𝑦 squared plus 𝐯 sub 𝑧 squared.

Squaring both sides of this equation gives us the magnitude of 𝐯 squared is equal to 𝐯 sub 𝑥 squared plus 𝐯 sub 𝑦 squared plus 𝐯 sub 𝑧 squared. Substituting this into our expression gives us the magnitude of 𝐯 squared over the magnitude of 𝐯 squared. We know that dividing any scalar by itself gives us an answer of one. This proves that cos squared 𝛼 plus cos squared 𝛽 plus cos squared 𝛾 is equal to one. We will now use this equation to help us solve a problem.

Suppose that 31 degrees, 65 degrees, and 𝜃 are the direction angles of a vector. Which of the following, to the nearest hundredth, is 𝜃? Is it (A) 72.88 degrees, (B) 84.00 degrees, (C) 85.03 degrees, or (D) 264.00 degrees?

We recall that if the three direction angles of a vector are 𝛼, 𝛽, and 𝛾, then cos squared 𝛼 plus cos squared 𝛽 plus cos squared 𝛾 is equal to one. In this question, we will substitute 𝛼 equal to 31 degrees, 𝛽 equal to 65 degrees, and 𝛾 equal to 𝜃. This gives us the equation shown. We recall that cos squared of 31 degrees is the same as cos of 31 degrees all squared. This is how we type this into a scientific calculator. cos squared 31 degrees plus cos squared 65 degrees is equal to 0.91334 and so on. We can then subtract this from both sides of our equation, giving us cos squared 𝜃 is equal to 0.08665 and so on.

We then square root both sides of this equation. cos 𝜃 is equal to 0.29437 and so on. Finally, we take the inverse cos of both sides, such that 𝜃 is equal to 72.8797 and so on. We want the answer to the nearest hundredth. This means that the nine in the thousandths column is the deciding number. Rounding up gives us 72.88 degrees. Therefore, the correct answer is option (A). The three direction angles of the vector are 61 degrees, 65 degrees, and 72.88 degrees.

In our next question, we need to work out the direction cosines of a vector.

Find the direction cosines of the vector 𝐀 with components five, two, and eight.

We recall that if vector 𝐯 has components 𝐯 sub 𝑥, 𝐯 sub 𝑦, and 𝐯 sub 𝑧 and direction angles 𝛼, 𝛽, and 𝛾, then the direction cosines cos of 𝛼, cos of 𝛽, and cos of 𝛾 are equal to 𝐯 sub 𝑥 over the magnitude of 𝐯, 𝐯 sub 𝑦 over the magnitude of 𝐯, and 𝐯 sub 𝑧 over the magnitude of 𝐯, respectively, where the magnitude of vector 𝐯 is equal to the square root of 𝐯 sub 𝑥 squared plus 𝐯 sub 𝑦 squared plus 𝐯 sub 𝑧 squared.

In this question, we are told that vector 𝐀 has components five, two, and eight. The magnitude of vector 𝐀 is therefore equal to the square roots of five squared plus two squared plus eight squared. Five squared is equal to 25, two squared is equal to four, and eight squared is equal to 64. Summing these three values gives us an answer of 93. Therefore, the magnitude of vector 𝐀 is the square root of 93. This means that the cos of angle 𝛼 is equal to five over the square root of 93, the cos of angle 𝛽 is equal to two over the square root of 93, and, finally, the cos of angle 𝛾 is equal to eight over the square root of 93. The direction cosines of vector 𝐀 are five over the square root of 93, two over the square root of 93, and eight over the square root of 93.

In our final question, we will need to calculate the direction angles from a geometric problem.

Find the measure of the direction angles of the vector 𝐅 represented by the given figure, correct to one decimal place.

We will begin by writing the vector 𝐅 in terms of its three components. In the 𝑥-direction, we travel eight centimeters. Therefore, the 𝑥-component of the vector is eight. In the 𝑦-direction, we travel 19 centimeters. Therefore, the 𝑦-component of vector 𝐅 is 19. We travel nine centimeters in the 𝑧- or 𝑧-direction. Therefore, the 𝑧-component is nine. Vector 𝐅 is equal to eight, 19, nine. The magnitude of vector 𝐅 is equal to the square root of eight squared plus 19 squared plus nine squared as we know that the magnitude of any vector is equal to the square root of the sum of the squares of the individual components.

Eight squared plus 19 squared plus nine squared is equal to 506. Therefore, the magnitude of vector 𝐅 is the square root of 506. We are asked to calculate the direction angles. These are usually labeled 𝛼, 𝛽, and 𝛾, where 𝛼 is the angle between the 𝑥-axis and the vector 𝐅, 𝛽 is the angle between the 𝑦-axis and the vector 𝐅, and, finally, 𝛾 is the angle between the 𝑧-axis and the vector 𝐅.

From our knowledge of the direction cosines, we know that 𝛼 is equal to the inverse cos of 𝐅 sub 𝑥 over the magnitude of 𝐅, 𝛽 is equal to the inverse cos of 𝐅 sub 𝑦 over the magnitude of 𝐅, and 𝛾 is equal to the inverse cos of 𝐅 sub 𝑧 over the magnitude of 𝐅, where 𝐅 sub 𝑥, 𝐅 sub 𝑦, and 𝐅 sub 𝑧 are the three components of vector 𝐅. We will now clear some space to calculate these values.

Angle 𝛼 is equal to the inverse cos of eight over the square root of 506. This is equal to 69.1671 and so on. Rounding to one decimal place, angle 𝛼 is equal to 69.2 degrees. 𝛽 is equal to the inverse cos of 19 over the square root of 506. This is equal to 32.3652 and so on. Rounding this to one decimal place gives us 32.4 degrees. Angle 𝛾 is equal to the inverse cos of nine over the square root of 506. This is equal to 66.4156 and so on. To one decimal place, angle 𝛾 is 66.4 degrees. The angles 𝛼, 𝛽, and 𝛾 can also be written as 𝜃 sub 𝑥, 𝜃 sub 𝑦, and 𝜃 sub 𝑧. In this question, they’re equal to 69.2 degrees, 32.4 degrees, and 66.4 degrees, respectively.

We will know summarize the key points from this video. The direction angles — often denoted 𝛼, 𝛽, and 𝛾 — are the angles between a vector and the 𝑥-, 𝑦-, and 𝑧- or 𝑧-axes, respectively. The direction cosines of any vector are such that the cos of angle 𝛼 is equal to 𝐯 sub 𝑥 divided by the magnitude of vector 𝐯, the cos of angle 𝛽 is equal to 𝐯 sub 𝑦 over the magnitude of vector 𝐯, and the cos of angle 𝛾 is equal to 𝐯 sub 𝑧 over the magnitude of vector 𝐯, where 𝐯 sub 𝑥, 𝐯 sub 𝑦, and 𝐯 sub 𝑧 are the components of the vector in the 𝑥-, 𝑦-, and 𝑧-directions.

The magnitude of a vector is equal to the square root of the sum of the squares of the individual components. We also proved in this video that the sum of the squares of the direction cosines is equal to one. cos squared 𝛼 plus cos squared 𝛽 plus cos squared 𝛾 is equal to one. We can use these equations to calculate the direction angles, direction cosines, or any other unknowns.