Question Video: Determining the Definition of the Direction Cosines | Nagwa Question Video: Determining the Definition of the Direction Cosines | Nagwa

Question Video: Determining the Definition of the Direction Cosines Mathematics • Third Year of Secondary School

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True or False: If (πœƒ_π‘₯, πœƒ_𝑦, πœƒ_𝑧) are defined as the direction angles of vector 𝐀, then (cos πœƒ_π‘₯, cos πœƒ_𝑦, cos πœƒ_𝑧) are defined as the direction cosines of vector 𝐀.

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Video Transcript

True or False: If πœƒ π‘₯, πœƒ 𝑦, and πœƒ 𝑧 are defined as the direction angles of vector 𝐀, then cos πœƒ π‘₯, cos πœƒ 𝑦, cos πœƒ 𝑧 are defined as the direction cosines of vector 𝐀.

We’re given that πœƒ π‘₯, πœƒ 𝑦, and πœƒ 𝑧 are the direction angles of a vector 𝐀. But what does this mean? Well, in general, given a vector 𝐀 with components 𝐴 π‘₯, 𝐴 𝑦, and 𝐴 𝑧, the direction angles of 𝐀 are the angles πœƒ π‘₯, πœƒ 𝑦, and πœƒ 𝑧 that the vector makes with the π‘₯-, 𝑦-, and 𝑧-axes, respectively. And these can be written as components πœƒ π‘₯, πœƒ 𝑦, and πœƒ 𝑧.

Now, we know that in a right angle trigonometry, the cos of an angle πœƒ is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. In our case, the length of the side adjacent to our angle πœƒ is the π‘₯-, 𝑦-, or 𝑧-component of our vector 𝐀, and the hypotenuse is the magnitude or norm of our vector 𝐀. And hence, the cos of our direction angle πœƒ π‘₯ is the π‘₯-component of our vector 𝐀 divided by the magnitude of 𝐀, and similarly, for the cos of direction angles πœƒ 𝑦 and πœƒ 𝑧. By definition then, the direction cosines of the vector 𝐀 are the cosines of the three direction angles πœƒ π‘₯, πœƒ 𝑦, and πœƒ 𝑧. So, cos πœƒ π‘₯, cos πœƒ 𝑦, and cos πœƒ 𝑧 are defined as the direction cosines of the vector 𝐀.

Therefore, the statement β€œIf πœƒ π‘₯, πœƒ 𝑦, πœƒ 𝑧 are defined as the direction angles of vector 𝐀, then cos πœƒ π‘₯, cos πœƒ 𝑦, cos πœƒ 𝑧 are defined as the direction cosines of vector 𝐀” is true.

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