Question Video: Finding the Measures of the Direction Angles of a Vector Mathematics

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

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Video Transcript

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 π‘₯-direction, we travel eight centimeters. Therefore, the π‘₯-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 π‘₯-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 π‘₯ 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 π‘₯, 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 πœƒ sub π‘₯, πœƒ sub 𝑦, and πœƒ sub 𝑧. In this question, they’re equal to 69.2 degrees, 32.4 degrees, and 66.4 degrees, respectively.

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