Cartesian Force Vectors
Any force F can be written in the form of a Cartesian vector. Such an expression uses the addition of the force’s component vectors in the x, y, and z directions of the axes of the right-hand coordinate. If the magnitude of its component in the x direction is Fx, in the y direction Fy, an in the z direction Fz, then F is expressed as
F = Fxi +Fyj + Fzk
wherei, j, and k are the Cartesian unit vectors.
Expressing A Force Vector By Using Its Magnitude And Two Points On Its Line Of Action. The Cartesian vector expression for a force is often obtained by using the Cartesian position vector that gives the Cartesian unit vectors corresponding to the direction concerned. For example, the force F has a line of action that passes through points P(x1,y1,z1) and Q(x2,y2,z2).
Let the position vector r that points from P to Q describes the relative position between P and Q. From the diagram, we obtain the relationship
r = –r1 + r2
However, we can formulate the relationships r1 = x1i + y1j + z1k and r2 = x2i + y2j + z2k. By inserting these relationships into the terms for r1 and r2 in the equation above, we get
r = (x2 – x1)i + (y2 – y1)j + (z2 – z1)k …..(2.2)
Hence the components of r in the x, y, and z directions are obtained by subtracting the coordinates of the tail from those of the heads. Note that if the coordinates of Q are subtracted from those of P, we will get the vector –r.
From Equation 2.22, we will get the unit vector s (corresponding to the line of action and the direction of F) as
s = r
In this expression, as usual, r is the magnitude of the position vector r and is given by the relationship
A force of magnitude F and having its line of action, direction, and sense corresponding to the unit vector s in Figure 2.3 can thus be expressed in the Cartesian vector form by writing F = Fs and then expanding the expression on the right of the equal sign.
The coordinate direction angles for the line of action of F are obtained from the cosine direction relationship for s, i.e.