Vector which has same angle with x,y,z axes

In summary, the conversation discusses finding a vector that forms the same angle with the three coordinate axes. The attempted solution of [1,1,1] is found to be incorrect and a new thread is created for a follow-up question. The correct unit vector is determined to be (\sqrt{3}/3, \sqrt{3}/3, \sqrt{3}/3).
  • #1
racnna
40
0

Homework Statement

?
How do i find a vector that has same angle with the three coordinate axes (x,y,z)?

The Attempt at a Solution



I immediately thought [1,1,1] would be it but it's not. I'm trying to find a plane whose normal vector forms the same angle with the three coordinate axes.
 
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  • #2
Why doesn't v=(1,1,1) work?

Let e,i,j stand for (1,0,0) ,(0,1,0) and (0,0,1) respectively.

Using a.b=|a||b|cost,

Then v.e=v.i=v.j =1 ; |e|=|i|=|j|=1 .Then

v.e/|v||e|= v.i/|v|.|i|= v.j/|v||j|=cost

Or did you have a different notion of angle in mind?
 
Last edited:
  • #3
hmm...it does work...

ok i need to ask a follow up question but i'll create a new thread
 
  • #4
If [itex]\alpha[/itex], [itex]\beta[/itex], and [itex]\gamma[/itex] are three angles, the unit vector that makes angle [itex]\alpha[/itex] with the x-axis, angle [itex]\beta[/itex] with the y-axis and [itex]\gamma[/itex] with the z-axis is [itex]cos(\alpha)\vec{i}+ cos(\beta)\vec{j}+ cos(\gamma)\vec{k}[/itex]. If all angles are the same, those three cosines are the same so any vector of the form (x, x, x), and in particular (1, 1, 1) will make equal angles with the three coordinate axes.

perhaps you are looking for the unit vector. The length of (x, x, x) is [itex]\sqrt{x^2+ x^2+ x^2}= x\sqrt{3}[/itex] and we want that equal to 1: we want [itex]x= 1/\sqrt{3}= \sqrt{3}/3[/itex]. The unit vector that makes equal angles with the coordinate axes is [itex](\sqrt{3}/3, \sqrt{3}/3, \sqrt{3}/3)[/itex].
 
  • #5
yes i got the same result for the unit vector. Thanks Ivy. Please see my other thread on the stress tensor. its a follow up to this question
 

1. What is a vector with the same angle with x, y, and z axes?

A vector with the same angle with x, y, and z axes is a vector that is oriented at the same angle with respect to all three axes. This means that the vector is pointing in the same direction as the axes, making it parallel to all three axes.

2. How do you determine the angle of a vector with the x, y, and z axes?

The angle between a vector and the x, y, and z axes can be determined by using trigonometric functions such as sine, cosine, and tangent. The angle can be calculated using the components of the vector (x, y, z) and the lengths of the axes.

3. Can a vector have different angles with the x, y, and z axes?

Yes, a vector can have different angles with the x, y, and z axes. The angle of a vector with each axis depends on its orientation and can vary depending on the direction of the vector. A vector can also have the same angle with two axes but a different angle with the third axis.

4. How does the angle of a vector with the x, y, and z axes affect its magnitude?

The angle of a vector with the x, y, and z axes does not affect its magnitude. The magnitude of a vector is only determined by its length and direction, not by its orientation with respect to the axes.

5. Can a vector with the same angle with x, y, and z axes have different magnitudes?

Yes, a vector with the same angle with x, y, and z axes can have different magnitudes. The angle of a vector only determines its direction, while its magnitude is determined by its length. Therefore, two vectors with the same angle but different lengths will have different magnitudes.

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