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_π₯, π_π¦, π_π§) are defined as the direction angles of vector π, then (cos π_π₯, cos π_π¦, cos π_π§) are defined as the direction cosines of vector π.

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### Video Transcript

True or False: If π π₯, π π¦, and π π§ are defined as the direction angles of vector π, then cos π π₯, cos π π¦, cos π π§ are defined as the direction cosines of vector π.

Weβre given that π π₯, π π¦, and π π§ are the direction angles of a vector π. But what does this mean? Well, in general, given a vector π with components π΄ π₯, π΄ π¦, and π΄ π§, the direction angles of π are the angles π π₯, π π¦, and π π§ that the vector makes with the π₯-, π¦-, and π§-axes, respectively. And these can be written as components π π₯, π π¦, and π π§.

Now, we know that in a right angle trigonometry, the cos of an angle π 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 is the π₯-, π¦-, or π§-component of our vector π, and the hypotenuse is the magnitude or norm of our vector π. And hence, the cos of our direction angle π π₯ is the π₯-component of our vector π divided by the magnitude of π, and similarly, for the cos of direction angles π π¦ and π π§. By definition then, the direction cosines of the vector π are the cosines of the three direction angles π π₯, π π¦, and π π§. So, cos π π₯, cos π π¦, and cos π π§ are defined as the direction cosines of the vector π.

Therefore, the statement βIf π π₯, π π¦, π π§ are defined as the direction angles of vector π, then cos π π₯, cos π π¦, cos π π§ are defined as the direction cosines of vector πβ is true.

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