How Do Dot Products Reflect Vector Projections?

Click For Summary
SUMMARY

The discussion centers on the mathematical concept of dot products in two-dimensional vectors, specifically how the dot product can be expressed both as the sum of the products of their components and as a product involving the cosine of the angle between them. The user highlights confusion regarding the scalar nature of projections and seeks clarity on how these two representations yield the same result. The concept of vector projections is emphasized, with a reference to Better Explained for further insights.

PREREQUISITES
  • Understanding of basic vector operations
  • Familiarity with trigonometric functions, specifically cosine
  • Knowledge of vector projections and their geometric interpretations
  • Basic grasp of linear algebra concepts
NEXT STEPS
  • Study the geometric interpretation of vector projections
  • Learn about the properties of dot products in higher dimensions
  • Explore applications of dot products in physics, particularly in work and energy calculations
  • Investigate the relationship between dot products and orthogonal vectors
USEFUL FOR

Students of mathematics, physics enthusiasts, and anyone seeking to deepen their understanding of vector calculus and linear algebra concepts.

Pochen Liu
Messages
52
Reaction score
2
I know that a dot product of 2, 2 dimension vectors a, b =

(ax * bx) + (ay * by)

but it also is equal to

a*bCos(θ)

because of "projections". That we are multiplying a vector by the 'scalar' property of the other vector which confuses me because that projection is in the direction of the other vector, therefore no scalar, so I cannot see how these two produce the same outcome.

What am I missing intuitively?
 
Last edited:
Physics news on Phys.org

Similar threads

How Do Four-Vectors with Orthogonal Dot Products Determine Each Other's Nature?
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Dot product between basis vectors and dual basis vectors
  • · Replies 17 ·
Replies
17
Views
2K
High School Confused about dot product multiplication
  • · Replies 33 ·
2
Replies
33
Views
4K
Undergrad Dot product of two vector operators in unusual coordinates
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Visualization of complex vectors and dot product
  • · Replies 14 ·
Replies
14
Views
4K
How do i choose the Surface area vector in different cases?
  • · Replies 11 ·
Replies
11
Views
2K
Vector dot product and parallel vectors
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Define Dot Product: A.B = ||A|| ||B|| cos(theta)
  • · Replies 4 ·
Replies
4
Views
3K
How Do Dot Product and Cross Product Differ in Vector Multiplication?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Inner and Outer Product of the Wavefunctions
  • · Replies 8 ·
Replies
8
Views
4K