How can A be expressed in terms of n as a unit vector?

In summary, the conversation discusses the use of the arbitrary vector A and unit vector n in a fixed direction to show that A can be expressed as (A.n).n + (A*n)*n. The conversation also mentions the use of Cartesian components and the notation for cross product.
  • #1
aigerimzh
15
0

Homework Statement


Let A be an arbitrary vector and let n be a unit vector in some fixed direction. Show that A=(A.n).n+(A*n)*n


Homework Equations





The Attempt at a Solution


I know that (A.n).n gives component of arbitrary vector, assume that it equals to Ax
 
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  • #2
aigerimzh said:

Homework Statement


Let A be an arbitrary vector and let n be a unit vector in some fixed direction. Show that A=(A.n).n+(A*n)*n


Homework Equations





The Attempt at a Solution


I know that (A.n).n gives component of arbitrary vector, assume that it equals to Ax

Most straightforward way is to write out the Cartesian components and verify. Just keep in mind that [itex]n_x^2 + n_y^2 + n_z^2 = 1[/itex].
 
  • #3
Again, you have used "*". What is that? The cross product? The usual notation is just "AX B".
 
  • #4
Yes, here also I mean (Axn)xn
 
  • #5
You can set up you own coordinate system and so, without loss of generality, take n to be [itex]\vec{i}[/itex]. Write A as [itex]a\vec{i}+ b\vec{j}+ c\vec{c}[/itex].

Then [itex]A\cdot n= a[/itex] so that [itex](A\cdot n)= a\vec{i}[/itex]. What are [itex]A\times n[/itex] and [itex](A\times n)\times n[/itex]?
 
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  • #6
I think that (Axn)xn= aj?
 
  • #7
aigerimzh said:
I think that (Axn)xn= aj?
No. Try again. What is Axn first?
 

1. What is an arbitrary vector?

An arbitrary vector is a mathematical concept that represents a quantity or magnitude that has both direction and magnitude. It is typically represented as a directed line segment with an arrow pointing in the direction of the vector.

2. How is an arbitrary vector different from a unit vector?

An arbitrary vector can have any magnitude and direction, while a unit vector has a magnitude of 1 and represents a specific direction in space. Unit vectors are often used to describe the orientation of a coordinate system or the direction of motion of an object.

3. What is the purpose of using unit vectors?

Unit vectors are useful because they allow us to express the direction of a vector without worrying about its magnitude. They also allow for easier calculations and comparisons between vectors, as they all have the same magnitude of 1.

4. How do you find the unit vector of an arbitrary vector?

To find the unit vector of an arbitrary vector, we divide the vector by its magnitude. This will result in a vector with the same direction, but a magnitude of 1.

5. Can an arbitrary vector be represented by more than one unit vector?

No, an arbitrary vector can only be represented by one unit vector. This is because unit vectors are unique and represent a specific direction in space. However, an arbitrary vector can be decomposed into multiple unit vectors in different directions, known as its vector components.

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