# What are direction cosines of a vector

## linkCatch

In vector calculus, the **Direction cosine** of a vector of Euclidean space \ ({\ displaystyle \ mathbb {R} ^ {3}} \) the cosine values of its direction angles, i.e. the angle between the vector and the three standard basis vectors \ ({\ displaystyle {\ vec {e}} _ {1}} \), \ ({\ displaystyle {\ vec {e}} _ {2}} \), \ ({\ displaystyle {\ vec {e}} _ {3}} \).^{[1]}^{[2]}

### properties

For the vector \ ({\ displaystyle {\ vec {v}} = {\ begin {pmatrix} v_ {1} \ v_ {2} \ v_ {3} \ end {pmatrix}}} \) are the direction cosines

- \ ({\ displaystyle \ cos \ alpha _ {1} = {\ frac {{\ vec {v}} \ cdot {\ vec {e}} _ {1}} {| {\ vec {v}} | \ , | {\ vec {e}} _ {1} |}} = {\ frac {v_ {1}} {| {\ vec {v}} |}} = {\ frac {v_ {1}} {\ sqrt {v_ {1} ^ {2} + v_ {2} ^ {2} + v_ {3} ^ {2}}}}} \),
- \ ({\ displaystyle \ cos \ alpha _ {2} = {\ frac {{\ vec {v}} \ cdot {\ vec {e}} _ {2}} {| {\ vec {v}} | \ , | {\ vec {e}} _ {2} |}} = {\ frac {v_ {2}} {| {\ vec {v}} |}} = {\ frac {v_ {2}} {\ sqrt {v_ {1} ^ {2} + v_ {2} ^ {2} + v_ {3} ^ {2}}}}} \),
- \ ({\ displaystyle \ cos \ alpha _ {3} = {\ frac {{\ vec {v}} \ cdot {\ vec {e}} _ {3}} {| {\ vec {v}} | \ , | {\ vec {e}} _ {3} |}} = {\ frac {v_ {3}} {| {\ vec {v}} |}} = {\ frac {v_ {3}} {\ sqrt {v_ {1} ^ {2} + v_ {2} ^ {2} + v_ {3} ^ {2}}}}} \),

as can also be read from the colored triangles in the adjacent figure. Conversely, \ ({\ displaystyle {\ vec {v}}} \) can be expressed by its absolute value and the direction cosines,

- \ ({\ displaystyle {\ vec {v}} = | {\ vec {v}} | {\ begin {pmatrix} \ cos \ alpha _ {1} \ cos \ alpha _ {2} \ cos \ alpha _ {3} \ end {pmatrix}}} \).

If this is divided by \ ({\ displaystyle | {\ vec {v}} |} \), it turns out that the direction cosine is precisely the components of the unit vector \ ({\ displaystyle {\ vec {e}} _ {v} } \) are in the direction of \ ({\ displaystyle {\ vec {v}}} \),

- \ ({\ displaystyle {\ vec {e}} _ {v} = {\ frac {\ vec {v}} {| {\ vec {v}} |}} = {\ begin {pmatrix} \ cos \ alpha _ {1} \ cos \ alpha _ {2} \ cos \ alpha _ {3} \ end {pmatrix}}} \).

Because of \ ({\ displaystyle | {\ vec {e}} _ {v} | = 1} \)

- \ ({\ displaystyle \ cos ^ {2} \ alpha _ {1} + \ cos ^ {2} \ alpha _ {2} + \ cos ^ {2} \ alpha _ {3} = 1} \).

Since the direction angles are limited to the range between \ ({\ displaystyle 0} \) and \ ({\ displaystyle \ pi} \) and the cosine is reversible in this interval, the three direction angles are given with the direction cosines.

### Individual evidence

- ↑ Gert Böhme: Introduction to higher mathematics (= Mathematics lectures for engineering schools. Volume 2). Springer, 1964, pp. 103-105 (restricted preview in Google Book Search).
- ↑ Eric W. Weisstein:
*Direction cosine*. In:*MathWorld*(English).

**Categories:**Analytical geometry

**Status of information:**11/24/2020 1:12:42 AM CET

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