Cartesian Vectors Mechanical Engineering Notes | EduRev

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Mechanical Engineering : Cartesian Vectors Mechanical Engineering Notes | EduRev

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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 = (x2x1)i + (y2y1)j + (z2z1)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

=                                                                    …..(2.3)

In this expression, as usual, r is the magnitude of the position vector r and is given by the relationship

r =

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 = Fand 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.

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