What is a mathematical explanation of the connection between: (1) projecting vector a onto vector b and multiplying the projected length of a with the length of vector b, and (2) the sum of the products of the equivalent components of the two vectors?

Lillie Pittman 2022-07-19 Answered
What is a mathematical explanation of the connection between: (1) projecting vector a onto vector b and multiplying the projected length of a with the length of vector b, and (2) the sum of the products of the equivalent components of the two vectors?
I realise there is a duality between a 2-dimensional vector and a 1x2 matrix, which can be used to explain the computation of the dot product. But I have not seen a satisfactory mathematical derivation, and was wondering whether there is another, simpler mathematical explanation.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

eishale2n
Answered 2022-07-20 Author has 15 answers
Let's start with the geometrical definition
a b = a b cos θ
Also, suppose that we have an orthonormal basis { e ^ i }. Then
a = i a i e ^ i b = i b i e ^ i
Now using the geometrical definition, if two of the basis vectors are the same
e ^ i e ^ i = e i e i cos 0 = 1
and if two vectors are different
e ^ i e ^ j = e i e j cos π 2 = 0
Then
a b = a ( i b i e ^ i ) = i ( a e ^ i ) b i = i a i b i

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-05-17
Find the scalar and vector projections of b onto a.
a=(4,7,4),b=(3,1,1)
asked 2021-02-11
Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in M2×4 with the property that FA = 0 (the zero matrix in M3×4). Determine if H is a subspace of M2×4
asked 2021-05-29
Find a vector equation and parametric equations for the line segment that joins P to Q.
P(0, - 1, 1), Q(1/2, 1/3, 1/4)
asked 2021-05-29
Which of the following expressions are meaningful? Which are meaningless? Explain.
a) (ab)c
(ab)c has ? because it is the dot product of ?.
b) (ab)c
(ab)c has ? because it is a scalar multiple of ?.
c) |a|(bc)
|a|(bc) has ? because it is the product of ?.
d) a(b+c)
a(b+c) has ? because it is the dot product of ?.
e) ab+c
ab+c has ? because it is the sum of ?.
f) |a|(b+c)
|a|(b+c) has ? because it is the dot product of ?.
asked 2022-08-17
Are two 2D vectors linearly dependent when x 1 y 2 = x 2 y 1 ?
I was working on a proof for why a set of three vectors R 2 are always linearly dependant when I came along this.
It appears that given two vectors [ x 1 , x 2 ] T and [ y 1 , y 2 ] T , if x 1 y 2 = x 2 y 1 then one vector is a scalar multiple of the other. This is the case in my proof where you have to divide by zero (more accurately, divide by x 1 y 2 x 2 y 1 ) so I feel like this can't be a coincidence.
An example is when x 1 y 2 = x 2 y 1 = 12, we could make pairs of vectors like v 1 = [ 12 , 4 ] T , v 2 = [ 3 , 1 ] T and then v 1 = 4 v 2
Does this hold for all vector pairs that satisfy the restriction and if so, why?
asked 2022-08-20
How to derive u T u v v T u u
asked 2020-11-02
Find the tangential and normal components of the acceleration vector r(t)=(3tt3)i+3t2j

New questions