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In the case of a vector, direction is one of the most important components. Without direction, you won't call it a vector, that would be a straight line that has magnitude only. We can find the direction of a vector with the help of cosine. In an orthonormal basis, direction cosines of the vector \vec { u } = (x, y, z) are:

 

\cos { \alpha } = \frac { x }{ \sqrt { { x }^{ 2 } + { y }^{ 2 } + { z }^{ 2 } }}

\cos { \beta } = \frac { y }{ \sqrt { { x }^{ 2 } + { y }^{ 2 } + { z }^{ 2 } }}

\cos { \gamma } = \frac { z }{ \sqrt { { x }^{ 2 } + { y }^{ 2 } + { z }^{ 2 } }}

\cos^{ 2 }{ \alpha } + \cos^{ 2 }{ \beta } + \cos^{ 2 }{ \gamma } = 1

 

It is necessary that you should be working on an orthonormal basis otherwise this formula won't be valid.

Example

Determine the direction cosines of the vector with components (1, 2, -3).

\cos { \alpha } = \frac { 1 }{ \sqrt { { 1 }^{ 2 } + { 2 }^{ 2 } + { (-3) }^{ 2 } } } = \frac { 1 }{ \sqrt { 14 } }

\cos { \beta } = \frac { 2 }{ \sqrt { { 1 }^{ 2 } + { 2 }^{ 2 } + { (-3) }^{ 2 } } } = \frac { 2 }{ \sqrt { 14 } }

\cos { \gamma } = \frac { -3 }{ \sqrt { { 1 }^{ 2 } + { 2 }^{ 2 } + { (-3) }^{ 2 } } } = \frac { -3 }{ \sqrt { 14 } }

 

{ (\frac { 1 }{ \sqrt { 14 } }) }^{ 2 } + { (\frac { 2 }{ \sqrt { 14 } }) }^{ 2 } + { (\frac { -3 }{ \sqrt { 14 } }) }^{ 2 } = \frac { 1 }{ 14 } + \frac { 4 }{ 14 } + \frac { 9 }{ 14 } = 1

 

A vector \vec { v } is drawn on the basis of orthonormal and has components (2, 5, -1). Find the direction cosines.

\cos { \alpha } = \frac { 2 }{ \sqrt { { 2 }^{ 2 } + { 5 }^{ 2 } + { (-1) }^{ 2 } } } = \frac { 2 }{ \sqrt { 30 } }

\cos { \beta } = \frac { 5 }{ \sqrt { { 2 }^{ 2 } + { 5 }^{ 2 } + { (-1) }^{ 2 } } } = \frac { 5 }{ \sqrt { 30 } }

\cos { \gamma } = \frac { -1 }{ \sqrt { { 2 }^{ 2 } + { 5 }^{ 2 } + { (-1) }^{ 2 } } } = \frac { -1 }{ \sqrt { 30 } }

 

{ (\frac { 2 }{ \sqrt { 30 } }) }^{ 2 } + { (\frac { 5 }{ \sqrt { 30 } }) }^{ 2 } + { (\frac { -1 }{ \sqrt { 30 } }) }^{ 2 } = \frac { 4 }{ 30 } + \frac { 25 }{ 30 } + \frac { 1 }{ 30 } = 1

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.