Diffrence between resolving vector to components and find projections

In summary, resolving a vector into components involves breaking it down into its orthogonal parts along specified axes, typically using trigonometric functions. This process helps understand the vector's influence in different directions. In contrast, finding projections involves determining how much of one vector lies in the direction of another, often using the dot product. While both concepts relate to vector decomposition, resolving components focuses on representation in terms of axes, whereas projections assess alignment and magnitude relative to another vector.
  • #1
Mohmmad Maaitah
90
20
Homework Statement
Resolve to components / Determine magnitude of projections
Relevant Equations
Dot product
I don't get what is the difference when I am asked to re-solve components and find projections to axes other than the Y and X
I know that the parallelogram works for the first one and the dot product for the second but what's the diffrence!
IMG_20230917_143410_282.jpg
1694950414012.jpg
 
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  • #2
Mohmmad Maaitah said:
Homework Statement: Resolve to components / Determine magnitude of projections
Relevant Equations: Dot product

I don't get what is the difference when I am asked to re-solve components and find projections to axes other than the Y and X
I know that the parallelogram works for the first one and the dot product for the second but what's the diffrence!
View attachment 332081View attachment 332082
If you resolve a vector ##\vec w## into components ##\vec u, \vec v## then ##\vec w=\vec u+\vec v##.
Those components will only be the projections of ##\vec w## onto ##\hat u, \hat v## if ##\vec u## and ##\vec v## are orthogonal.

Writing ##\vec u=u\hat u## etc. and ##\lambda=\hat u\cdot\hat v##,
##\vec w=u\hat u+v\hat v##
##\vec w\cdot\hat u=u+v\lambda##
etc., whence
##v=\frac{\vec w\cdot\hat v-\vec w\cdot\hat u\lambda}{1-\lambda^2}##.
 
Last edited:
  • #3
haruspex said:
If you resolve a vector ##\vec w## into components ##\vec u, \vec v## then ##\vec w=\vec u+\vec v##.
Those components will only be the projections of ##\vec w## onto ##\hat u, \hat v## if ##\vec u## and ##\vec v## are orthogonal.

Writing ##\vec u=u\hat u## etc. and ##\lambda=\hat u\cdot\hat v##,
##\vec w=u\hat u+v\hat v##
##\vec w\cdot\hat u=u+v\lambda##
etc., whence
I still don't get the diffrence between projection and force component.
 
  • #4
Mohmmad Maaitah said:
I still don't get the diffrence between projection and force component.
The projection of one vector on another depends only on those two vectors. It is unaffected by any other vectors under consideration.
If you are resolving into components then you need a set of directions to resolve into, ##\hat u_i##, and the coefficient to use in one direction depends on the whole set of directions. If you modify ##\hat u_1## then you may find the magnitude of the component in the ##\hat u_2## direction changes.
 
  • #5
I get it thank you sir!
haruspex said:
The projection of one vector on another depends only on those two vectors. It is unaffected by any other vectors under consideration.
If you are resolving into components then you need a set of directions to resolve into, ##\hat u_i##, and the coefficient to use in one direction depends on the whole set of directions. If you modify ##\hat u_1## then you may find the magnitude of the component in the ##\hat u_2## direction changes.
 

FAQ: Diffrence between resolving vector to components and find projections

What is the main difference between resolving a vector into components and finding projections?

Resolving a vector into components involves breaking down a vector into its perpendicular components, typically along the x and y axes in a 2D plane, or x, y, and z axes in a 3D space. Finding the projection of a vector involves projecting one vector onto another, which results in a new vector that lies along the line of the second vector, showing how much of the first vector lies in the direction of the second vector.

How do you resolve a vector into its components?

To resolve a vector into its components, you use trigonometric functions based on the vector's angle with the coordinate axes. For example, in 2D, if you have a vector \( \vec{A} \) with magnitude \( A \) and angle \( \theta \) from the x-axis, its components are \( A_x = A \cos(\theta) \) and \( A_y = A \sin(\theta) \). In 3D, you would also consider the z-component.

What is the formula for finding the projection of one vector onto another?

The projection of vector \( \vec{A} \) onto vector \( \vec{B} \) is given by the formula \( \text{proj}_{\vec{B}} \vec{A} = \left(\frac{\vec{A} \cdot \vec{B}}{\vec{B} \cdot \vec{B}}\right) \vec{B} \), where \( \vec{A} \cdot \vec{B} \) is the dot product of the vectors and \( \vec{B} \cdot \vec{B} \) is the magnitude squared of \( \vec{B} \).

In what scenarios would you use vector components versus vector projections?

Vector components are often used in physics and engineering to simplify the analysis of forces, motion, and other vector quantities by breaking them down into perpendicular directions. Vector projections are used to determine how much of one vector lies in the direction of another, which is useful in various applications like finding the work done by a force, in computer graphics for shading calculations, and in vector decomposition.

Can the projection of a vector result in a vector with a different direction than the original vector?

No, the projection of a vector \( \vec{A} \) onto another vector \( \vec{B} \) will always result in a vector that is parallel to \( \vec{B} \). The direction of the projection vector is along \( \vec{B} \), showing how much of \( \vec{A} \) aligns with \( \vec

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