Dot product, is (a.b).(a.b)=(a.a).(b.b) ?

Click For Summary
SUMMARY

The discussion clarifies that the equation (a.b)(a.b) = (a.a)(b.b) is incorrect. The dot product is commutative, meaning (a.b) = (b.a), but this does not imply that the products of the dot products are equal. The user provided vectors a = (2, 2, 0) and b = (1, 0, 0) to illustrate the discrepancy, where (a.b)(a.b) results in 4 while (a.a)(b.b) results in 8. The notation used in the original equation was misleading, as it conflated the dot product with standard multiplication.

PREREQUISITES
  • Understanding of vector operations, specifically dot products.
  • Familiarity with vector notation and mathematical symbols.
  • Basic knowledge of commutative properties in mathematics.
  • Ability to interpret and manipulate mathematical expressions accurately.
NEXT STEPS
  • Study the properties of dot products in vector algebra.
  • Learn about vector notation and how to properly represent mathematical operations.
  • Explore LaTeX for typesetting mathematical expressions effectively.
  • Investigate common misconceptions in vector mathematics and their resolutions.
USEFUL FOR

Students, educators, and professionals in mathematics, physics, and engineering who seek to deepen their understanding of vector operations and clarify common misconceptions regarding dot products.

lemd
Messages
31
Reaction score
0
Hi,

As I remember, dot product is commutative, and so (a.b).(a.b) = (a.a).(b.b)
But when I apply to simple vectors it is all wrong, e.g:
a = (2, 2, 0)
b = (1, 0, 0)

(a.b).(a.b) = 2.2 = 4
(a.a).(b.b) = 8.1 = 8

Why are they different? Pls explain for me

Thanks
 
Physics news on Phys.org
That is not what "commutative" means. It meas that (a.b) = (b.a)

It should be obvious that (a.b).(a.b) is not equal to (a.a).(b.b) in general. For example (a.b) can be 0 when (a.a) and (b.b) are both greater than 0.
 
Also, your notation is misleading -- "(a.b).(a.b)". You are using a period (.) to indicate two different types of multiplication: the dot or scalar product that is defined for two vectors, and ordinary multiplication of real numbers.

Slightly better notation would be (a.b)(a.b), with nothing shown for the real number multiplication.

Even better would be to use a dot for the dot product, for which some simple LaTeX can be used: (a ##\cdot## b)(a ##\cdot## b).
 
Many thanks
 

Similar threads

Undergrad Orthogonal transformations preserve length
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Why is A.A = ||A||^2 in Vector Analysis?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Why does A.A = ||A||^2 in the scalar product formula?
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Scalar product and vector product
  • · Replies 5 ·
Replies
5
Views
2K
High School Confused about dot product multiplication
  • · Replies 33 ·
2
Replies
33
Views
5K
Undergrad Dot product, inner product, and projections
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Confused about dot product of a and b = |a||b| if theta = 0
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Define Dot Product: A.B = ||A|| ||B|| cos(theta)
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Dot product in Euclidean Space
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Visualization of complex vectors and dot product
  • · Replies 14 ·
Replies
14
Views
4K