Law of Cosines in Linear Algebra: Understanding the Dot Product of Unit Vectors

[TGV320]

TL;DR Summary

Why cosθ？

HI,

I am studying linear algebra, and I just can't understand why "Unit vectors u and U at angle θ have u multiplied by U=cosθ

Why is it like that?

Thanks

 

[martinbn]

Because that is the definition of dot product. It is the product of the lengths and the cos of the angle. If the vectors are unit, the lengths are both 1.

 

[BvU]

@TGV320 : can you explain the question? To me it isn't clear what specifically you don't understand.

Do you have the same difficulty with projection of a vector on another ? with coponents in a coordinate system ?

$\$

 

[DrClaude]

There is a picture in this Wikipedia article that should clarify everything:
https://en.wikipedia.org/wiki/Scalar_projection

 

[mathman]

For 2-d vectors $a=(a_1,a_2),(b_1,b_2)$, dot product =$(a_1b_1+a_2b_2)$.. Work out trig. to get angle.

 

[Mark44]

For 2-d vectors $a=(a_1,a_2),(b_1,b_2)$, dot product =$(a_1b_1+a_2b_2)$.. Work out trig. to get angle.

There are two definitions of the dot product for 2D vectors:
Coordinate definition, as you wrote.
Coordinate-free definition: $\vec a \cdot \vec b = |\vec a||\vec b|\cos(\theta)$, where $\theta$ is the smaller of the angles between the two vectors.

 

[mathman]

There are two definitions of the dot product for 2D vectors:
Coordinate definition, as you wrote.
Coordinate-free definition: $\vec a \cdot \vec b = |\vec a||\vec b|\cos(\theta)$, where $\theta$ is the smaller of the angles between the two vectors.

I prefer the first, since we don't know the angle.

 

[Mark44]

I prefer the first, since we don't know the angle.

Each definition has its uses. For example, if you know the value of the dot product, and the magnitudes of the vectors, but don't know the coordinates of the vectors, you can use the coordinate-free definition to calculate the angle.

With regard to unit vectors, the subject of this thread, if you know the value of their dot product, you calculate the angle between them.

I've seen many problems where the coordinate definition could not be used.

 

[BvU]

@TGV320 ?

 

[TGV320]

Thanks, I think I have a better understanding now.
Never learned that before at school, confused me quite a lot the first time.

 

[mathwonk]

the fact that the two versions are the same is called the law of cosines. perhaps you learned it in that form in trig.