How to Resolve a Vector into Parallel and Perpendicular Components?

In summary, dot product is a mathematical operation that represents the scalar projection of one vector onto another. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. This value represents the component of the first vector in the direction of the second vector.
  • #1
songoku
2,266
319

Homework Statement


Resolve the vector 6i+2j-2k into two vectors, one parallel and another perpendicular to i+j+k


Homework Equations


[tex]a\cdot b = 0 \; \text{for two perpendicular vectors}[/tex]

[tex]a=\lambda \;b\;\text{for parallel vectors}[/tex] , [tex]\lambda = \text{parameter}[/tex]


The Attempt at a Solution


I have no idea to start. How to resolve one component of vector into two components ?

Thanks
 
Physics news on Phys.org
  • #2
Have you learned about the geometrical meaning of dot product?
I.e. if I drew v = 6i+2j-2k and w = i+j+k for you, could you explain to me how v · w appears in the picture?
 
  • #3
CompuChip said:
Have you learned about the geometrical meaning of dot product?
I.e. if I drew v = 6i+2j-2k and w = i+j+k for you, could you explain to me how v · w appears in the picture?

I don't think that he can give you geometrical interpretation because dot product is just a scalar and not vector.
Although, he can give you the scalar projection of v onto w.

And do you mean to resolve 6i+2j-2k = c + d ?

If so, let c be the vector parallel to and d perpendicular to the vector (1,1,1) i.e i+j+k

(cx,cy,cz)=[itex]\lambda[/itex](1,1,1)

and do you know what d will equal to? What are the conditions of the task?Regards.
 
Last edited:
  • #4
songoku said:

Homework Statement


Resolve the vector 6i+2j-2k into two vectors, one parallel and another perpendicular to i+j+k


Homework Equations


[tex]a\cdot b = 0 \; \text{for two perpendicular vectors}[/tex]

[tex]a=\lambda \;b\;\text{for parallel vectors}[/tex] , [tex]\lambda = \text{parameter}[/tex]


The Attempt at a Solution


I have no idea to start. How to resolve one component of vector into two components ?

Thanks
Okay, so a vector parallel to i+ j+ k must be [itex]\lambda i+ \lambda j+ \lambda k[/itex]. Suppose ai+ bj+ ck is the vector perpendicular to that. Then you have [itex](ai+ bj+ ck)\cdot(\lamba i+ \lambda j+ \lambda k)= a\lambda+ b\lambda+ c\lambda[/itex][itex]= \lambda(a+ b+ c)= 0[/itex] and [itex](ai+ bj+ ck)+ (\lambda i+ \lambda j+ \lambda k)[/itex][itex]= (a+\lambda)i+ (b+\lambda)j+ (c+ \lambda)k[/itex][itex]= 6i+ 2j- 2k[/itex]. That gives you four equations to solve for a, b, c, and [itex]\lambda[/itex].
 
  • #5
Hi CompuChip, Дьявол, and Mr. HallsofIvy

I get it now. Sorry, but I have another simple question. What is the meaning of dot product ?

It's easier for me to imagine cross product. If we cross two vectors, we will get third vector that is perpendicular to the previous two vectors.

But, what about dot product ? If we dot 2 vectors, we get a numerical value, What does the numerical value represent?

Thanks
 
  • #6
As I was taught, the dot product is where you only consider the part of the second vector being multiplied that is parallel to the first vector.
 
  • #7
songoku said:
Hi CompuChip, Дьявол, and Mr. HallsofIvy

I get it now. Sorry, but I have another simple question. What is the meaning of dot product ?

It's easier for me to imagine cross product. If we cross two vectors, we will get third vector that is perpendicular to the previous two vectors.

But, what about dot product ? If we dot 2 vectors, we get a numerical value, What does the numerical value represent?

Thanks

http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Dot_Product.svg/300px-Dot_Product.png [Broken]

And the dot product is A • B = |A| cos(θ) |B|

|A| cos(θ) is the scalar projection of A onto B.

So you got the part down there, just you multiply it with the magnitude of B.

Regards.
 
Last edited by a moderator:
  • #8
Hi Дьявол and mg0stisha

Wow, now I get the meaning of dot product. Thanks a lot to all of you ! (CompuChip, Дьявол, MR. HallsofIvy, mg0stisha) ^^
 

1. What is a vector?

A vector is a mathematical representation of a quantity that has both magnitude and direction.

2. Why do we need to resolve a vector into parallel and perpendicular components?

Resolving a vector into parallel and perpendicular components allows us to break down a vector into its individual parts, making it easier to analyze and calculate.

3. How do you calculate the parallel component of a vector?

To calculate the parallel component of a vector, you can use the formula P = V * cosθ, where P is the parallel component, V is the magnitude of the vector, and θ is the angle between the vector and the direction you want the parallel component to point.

4. How do you calculate the perpendicular component of a vector?

To calculate the perpendicular component of a vector, you can use the formula Q = V * sinθ, where Q is the perpendicular component, V is the magnitude of the vector, and θ is the angle between the vector and the direction you want the perpendicular component to point.

5. Can a vector have more than two components?

Yes, a vector can have any number of components, depending on the number of dimensions in which it is being represented. For example, a vector in three-dimensional space would have three components, while a vector in two-dimensional space would have two components.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
20
Views
675
  • Precalculus Mathematics Homework Help
Replies
8
Views
1K
  • Precalculus Mathematics Homework Help
Replies
9
Views
2K
  • Precalculus Mathematics Homework Help
Replies
11
Views
2K
  • Precalculus Mathematics Homework Help
Replies
21
Views
7K
  • Precalculus Mathematics Homework Help
Replies
17
Views
13K
  • Precalculus Mathematics Homework Help
Replies
20
Views
2K
  • Precalculus Mathematics Homework Help
Replies
6
Views
3K
  • Precalculus Mathematics Homework Help
Replies
3
Views
2K
  • Precalculus Mathematics Homework Help
Replies
12
Views
3K
Back
Top