Question

Define dierection cosines of a vector

Answer 1

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The direction cosines are the components of the unit vector in the same direction as the original vector. Don’t miss the unit vectorpart!

So given that A⃗ =Axa^x+Aya^y+Aza^zA→=Axa^x+Aya^y+Aza^z, to determine the direction cosines, first calculate the unit vector.

A^=Ax|A⃗ |a^x+Ay|A⃗ |a^y+Az|A⃗ |a^zA^=Ax|A→|a^x+Ay|A→|a^y+Az|A→|a^z

The direction cosines are the components of A^A^.

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Answer 2

The direction cosine of vector:

Definition.

The direction cosines of the vector a are the cosines of angles that the vector forms with the coordinate axes.

The direction cosines uniquely set the direction of vector.

Basic relation. To find the direction cosines of the vector a is need to divided the corresponding coordinate of vector by the length of the vector.

The coordinates of the unit vector is equal to its direction cosines.
Property of direction cosines. The sum of the squares of the direction cosines is equal to one.
Direction cosines of a vector formulas
Direction cosines of a vector formula for two-dimensional vector
In the case of the plane problem (Fig. 1) the direction cosines of a vector a = {ax ; ay} can be found using the following formula

cos α = ax ; cos β = ay

|a| |a|


Property:

cos2 α + cos2 β = 1

Direction cosines of a vector 2d

Direction cosines of a vector formula for three-dimensional vector

In the case of the spatial problem (Fig. 2) the direction cosines of a vector a = {ax ; ay ; az} can be found using the following formula

cos α = ax ; cos β = ay ; cos γ = az

|a| |a| |a|

Property:

cos2 α + cos2 β + cos2 γ = 1

Direction cosines of a vector 3d
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