3-D Vector Components   


3-D Vector Components

Recall, from the previous section, a vector in the x-y plane in can be written in Cartesian notation as,

     F = F_xi + F_yj

Here i and j are unit vectors in the x and y directions. Likewise, a vector can be written in the three dimensional x-y-z space in Cartesian form,

     F = F_xi + F_yj + F_zk

The new component, k is the unit vector in the z direction. The components F_x, F_y, and F_z can be determined from the magnitude and direction of F,

     F_x = F cosθ_x
     F_y = F cosθ_y
     F_z = F cosθ_z

The angle θ_i is the angle that F makes with the i axis as shown in the diagram at the top left.

The order of the x-y-z coordinates does matter. If the three coordinates are aligned so that the thumb is in the x-direction when the first finger is in the y-direction and the second finger is in the z-direction, then it is called a right-handed coordinate system. Otherwise, it would be a left-handed coordinate system. Engineering always uses right-handed coordinate system. There is nothing wrong with a left-handed system, but all books, equations and technical papers use a right-handed system for convention.

Direction Cosines

Addition of 3-D vectors is the same as for 2-D except that three components must be added instead of two. If two vectors F₁ and F₂ exist, then the addition of F₁ and F₂ is,

     F_R = F₁+ F₂

     F_R = (F_1x + F_2x)i + (F_1y + F_2y)j + (F_1z + F_2z)k

The magnitude of the resultant vector is determined by applying the Pythagorean theorem,

     F_R = ( F_Rx² + F_Ry² + F_Rz² )^0.5

The direction cosines are useful in some problems to identify each direction component. They are given as,

     cosθ_x = F_Rx /F_R
     cosθ_y = F_Ry /F_R
     cosθ_z = F_Rz /F_R

General unit vectors can be determined in 3D using the definition of unit vectors, as

     u_F = F/F

Substituting vector notation and magnitude for F, gives

This relationship is useful when the direction of the vector is needed.

In statics, force and moment vectors are the most commonly but many times a position vector is needed to help determine the direction of the force or moment vector. Position vectors are the same as all vectors, but they describe direction and distance,

     r = xi + yj + zk

where x, y, and z are distances (scalars). Generally, position vectors are determined by its two end points, giving

     r = (x_B - x_A)i + (y_B - y_A)j + (z_B - z_A)k

The directional unit vector for a position vector is

   u_r = r/|r
   

Spherical Angles Diagram  

Spherical Coordinate Angles

If the direction of the vector is defined by the two angles θ and φ, (shown at the left) as is commonly done in spherical coordinate systems, then the components are given by

     F_x = F sinφ cosθ
     F_y = F sinφ sinθ
     F_z = F cosφ

The spherical angles θ and φ are given by,

     φ = cos^-1 F_Rz /F_R

     θ = sin^-1 F_Ry /(F_R sinφ)
      = cos^-1 F_Rx /(F_R sinφ)