*I suggest that you read* post 17.2 *before this*.

In post 17.2 we saw how to add two-dimensional vectors – vectors that could be drawn on a sheet of paper. But forces can act in three dimensions – up-and-down, as well as north-south and east-west. A three-dimensional force has the same relationship to a two-dimensional force as a sculpture has to a picture.

How do we do vector algebra (see post 17.2) in three-dimensions? We start off by defining a three-dimensional orthogonal Cartesian coordinate system; we met a system like this, but in two dimensions, in appendix 2 of post 16.50. Now we have an *X*-axis and a *Y*-axis, that are perpendicular to each other and cross at an origin. But we add a third axis, the *Z*-axis, that is perpendicular to both the *X*-axis and the *Y*-axis. When we add the *Z*-axis, we need to be careful because three-dimensional Cartesian coordinate systems can be either right-handed or left-handed.

We usually construct right-handed coordinate systems. To do this, curl the fingers of your right hand – so your fingers point from the *X*-axis to the *Y*-axis. If you point your thumb up, it then points in the *Z*-axis direction; in the picture on the right-hand side, the *Z*-axis points towards you, out of the plane of the paper.

In the same way as we defined unit vectors ** i** and

**in the**

*j**X*and

*Y*-axes directions (post 17.2), we now define a unit vector

**in the**

*k**Z*-axis direction. Remember that a unit vector simply defines a direction and has a length equal to the number one and no units (like metres or newtons).

So we can write the vector representing a three-dimensional force as

** F** =

*i**F*+

_{x}

*j**F*+

_{y}

*k**F*.

_{z}Here *F _{x}*,

*F*and

_{y}*F*are the components of

_{z}**in the**

*F**X*,

*Y*and

*Z*-axis directions. The magnitude of

**is given, by Pythagoras’ theorem (see the appendix at the end of this post for more explanation), by**

*F**F*^{2} = *F _{x}*

^{2}+

*F*

_{y}^{2}+

*F*

_{z}^{2}so that

*F*= √(

*F*

_{x}^{2}+

*F*

_{y}^{2}+

*F*

_{z}^{2}).

There are two related ways in which we can use angles to specify the direction of ** F**. We will use a simple extension of the method we used for two-dimensional vectors by defining the angle

*φ*that

**makes with the**

*F**Z*-axis. Now we define

*θ*as the angle between the projection of

**on to the**

*F**XY*-plane and the

*X*-axis (see picture above). From the definitions of cosine and sine (appendix 3 of post 16.50)

*F _{x}* =

*F*sin

*φ*cos

*θ*,

*F*=

_{y}*F*sin

*φ*sin

*θ*and

*F*=

_{z}*F*cos

*φ*.

We can calculate the angles *θ* and *φ* from the components of the vectors. The angle *θ* is calculated in exactly the same way as for a two-dimensional vector (equation 5 of post 17.2). The angle *φ* is calculated from

*φ* = arccos(*F _{z}*/

*F*)

where arccos(*x*) just means “the angle whose cosine is *x*”. You can look up arccos values at: http://www.rapidtables.com/calc/math/Arccos_Calculator.htm.

The resultant (sum) of two vectors

** F** =

*i**F*+

_{x}

*j**F*+

_{y}

*k**F*and

_{z}**=**

*f*

*i**f*+

_{x}

*j**f*+

_{y}

*k**f*.

_{z}is then given by

** R** =

**+**

*F***=**

*f***(**

*i**F*+

_{x}*f*) +

_{x}**(**

*j**F*+

_{y}*f*) +

_{y}**(**

*k**F*+

_{z}*f*).

_{z}So, the algebra of three-dimensional vectors is a simple extension of the algebra of two-dimensional vectors. We can even extend the idea of a vector into 4 or more dimensions; but it is then easier to represent vectors using the notation of matrix algebra which is a bit more difficult to explain.

*Related posts*

17.2 Vectors

16.50 Direction of forces

*Follow-up posts*

*Appendix*

The purpose of this appendix is to show that *F*^{2} = *F _{x}*

^{2}+

*F*

_{y}^{2}+

*F*

_{z}^{2}.

In the figure, the brown line is the projection (see appendix 3 of post 16.50) of the vector ** F** on to the

*XY*-plane.

According to Pythagoras’ theorem, the length of this line is √(*F _{x}*

^{2}+

*F*

_{y}^{2}). Here

*F*is the length of the blue line and

_{x}*F*is the length of the red line. The statements in this paragraph are explained and proved in post 17.2.

_{y}The green line is the projection of ** F **on to the

*Z*-axis and is denoted by

*F*; the dashed green line is parallel to it and has the same length.

_{z}** F**, the dashed green line and the brown line form a right-angled triangle. So, by Pythagoras’ theorem

*F*^{2} = (√(*F _{x}*

^{2}+

*F*

_{y}^{2}))

^{2}+

*F*

_{z}^{2}=

*F*

_{x}^{2}+

*F*

_{y}^{2}+

*F*

_{z}^{2}.