In the vector calculation are direction cosines of a vector of the Euclidean space , the cosine of its directional angle, ie the angle between the vector and the three standard basis vectors , , .
R.
3
{\ displaystyle \ mathbb {R} ^ {3}}
e
→
1
{\ displaystyle {\ vec {e}} _ {1}}
e
→
2
{\ displaystyle {\ vec {e}} _ {2}}
e
→
3
{\ displaystyle {\ vec {e}} _ {3}}
properties
For the vector , the direction cosines are
v
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=
(
v
1
v
2
v
3
)
{\ displaystyle {\ vec {v}} = {\ begin {pmatrix} v_ {1} \\ v_ {2} \\ v_ {3} \ end {pmatrix}}}
cos
α
1
=
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→
⋅
e
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1
|
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|
|
e
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1
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=
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1
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=
v
1
v
1
2
+
v
2
2
+
v
3
2
{\ 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}}}}}
,
cos
α
2
=
v
→
⋅
e
→
2
|
v
→
|
|
e
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2
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=
v
2
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=
v
2
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}}}}}
,
cos
α
3
=
v
→
⋅
e
→
3
|
v
→
|
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e
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3
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=
v
3
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=
v
3
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, it can be expressed by its absolute value and the direction cosine,
v
→
{\ displaystyle {\ vec {v}}}
v
→
=
|
v
→
|
(
cos
α
1
cos
α
2
cos
α
3
)
{\ displaystyle {\ vec {v}} = | {\ vec {v}} | {\ begin {pmatrix} \ cos \ alpha _ {1} \\\ cos \ alpha _ {2} \\\ cos \ alpha _ {3} \ end {pmatrix}}}
.
If this is divided by, it turns out that the direction cosines are just the components of the unit vector in the direction of ,
|
v
→
|
{\ displaystyle | {\ vec {v}} |}
e
→
v
{\ displaystyle {\ vec {e}} _ {v}}
v
→
{\ displaystyle {\ vec {v}}}
e
→
v
=
v
→
|
v
→
|
=
(
cos
α
1
cos
α
2
cos
α
3
)
{\ displaystyle {\ vec {e}} _ {v} = {\ frac {\ vec {v}} {| {\ vec {v}} |}} = {\ begin {pmatrix} \ cos \ alpha _ { 1} \\\ cos \ alpha _ {2} \\\ cos \ alpha _ {3} \ end {pmatrix}}}
.
Because is
|
e
→
v
|
=
1
{\ displaystyle | {\ vec {e}} _ {v} | = 1}
cos
2
α
1
+
cos
2
α
2
+
cos
2
α
3
=
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 and and the cosine is reversible in this interval, the three direction angles are also given with the direction cosines.
0
{\ displaystyle 0}
π
{\ displaystyle \ pi}
Individual evidence
^ Gert Böhme : Introduction to higher mathematics (= mathematics - lectures for engineering schools . Volume 2 ). Springer, 1964, p. 103–105 ( limited preview in Google Book search).
↑ Eric W. Weisstein : Direction Cosine . In: MathWorld (English).
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