2
$\begingroup$

Let's imagine a point in 3D system that its distance of that point to the origin 1 unit.

If the coordinates of that points have been given as x = a, y = b, z = c.

How can we calculate three angles to xyz axises of a vector from origin to that point ?

$\endgroup$
  • $\begingroup$ Hint: dot product relates to the cosine of an angle. $\endgroup$ – orion Apr 14 '18 at 7:38
6
$\begingroup$

The vector point coordinates are $OP=(a,b,c)$ then the angles with $x,y,z$ with unitary vectors $e_1=(1,0,0),e_2=(0,1,0),e_3(0,0,1)$ are given by the dot product

  • $\cos \alpha = \frac{OP\cdot e_1}{|OP||e_1|}=OP\cdot e_1=a$
  • $\cos \beta = \frac{OP\cdot e_2}{|OP||e_2|}=OP\cdot e_2=b$
  • $\cos \gamma = \frac{OP\cdot e_3}{|OP||e_3|}=OP\cdot e_3=c$
$\endgroup$
1
$\begingroup$

I used a way finding the angle between two vectors.

angle = aCos ( (V1.V2) / (|V1|.|V2|)  )

When I want to calculate the angle of a point to x axis.

x Axis is

V1 = 1.i + 0.j + 0.k  

The point is

V2 = a.i + b.j + c.k

This way I calculated correct solution.

$\endgroup$
  • $\begingroup$ Looks the same with @gimusi 's answer. $\endgroup$ – Hope Apr 14 '18 at 8:16
  • $\begingroup$ yes for $P(a,b,c)$ with $a^2+b^2+c^2=1$ $a,b,c$ represent the cos of the angles with the axes. $\endgroup$ – user Apr 14 '18 at 8:18
1
$\begingroup$

Suppose you have a vector $\vec v = xi+yj+zk$ where $i,j,k $ are the basis unit vectors then the angles $\alpha,\beta, \gamma$ of the vector to the $x,y,z $ axes respectively is given by ;

$\alpha = \frac{x}{\sqrt{x^2+y^2+z^2}} = \cos(a)\\\beta = \frac{y}{\sqrt{x^2+y^2+z^2}}=\cos(b)\\\gamma = \frac{z}{\sqrt{x^2+y^2+z^2}}=\cos(c)$

It follows that squaring the 3 equations and adding them results in

$\cos^2(a)+\cos^2(b)+\cos^2(c) =\alpha^2+\beta^2+\gamma^2 = 1 $

enter image description here

$\endgroup$
1
$\begingroup$

Suppose angle of vector related to x axis is $\alpha$, related to y axis is $\beta$ and related to z axis is $\gamma$ then we have:

Due to presumption; $\sqrt {a^2+b^2+c^2}=1$

$1\times \ cos \alpha=a$

$1\times \ cos \beta=b$

$1 \times \ cos \gamma=c$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.