3.2: Drawing vectors
Although the motion of objects, can be fully expressed with mathematical equations, it is often much easier to visualise objects and mechanisms using drawings. Moreover, it is common practice in engineering to not provide all information in equation form, but provide part of it in the form of a drawing. Therefore, correctly drawing vectors, and interpreting those drawings is important for a correct analysis of dynamics. Analysing a problem in mechanics usually starts by making a sketch of the system, and later one or more free-body diagrams (FBDs), which are then translated into mathematical equations. To make sure no errors or inconsistencies occur during this translation process from drawing to mathematical equations, it is important to draw all vectors correctly and to have clear definitions on the way vectors are drawn and converted into equations.
A vector is always drawn as a straight arrow, with a label. Optionally, a dot can be placed at the tail or head of the arrow to indicate the point of action of the vector. If the arrow points perpendicular to the plane of the drawing,
its direction can be indicated by the symbols \(\odot\) (pointing towards the reader out of the page) and \(\otimes\) (pointing away from the reader into the page). The meaning of these symbols can be easily memorised by imagining a dart arrow that points in the same direction as the vector and realising that the symbols resemble the tip \((\odot)\), or flight \((\otimes)\) of the dart arrow.
Figure 3.2 shows three main ways to specify vectors in a drawing:
- Drawing general vectors \(\overrightarrow{\boldsymbol{F}}_{A}, \overrightarrow{\boldsymbol{F}}_{B}\) (Figure 3.2a).
- Drawing vectors as an arrow with fixed direction and scalar signed magnitude \(F_{A}, F_{B}\) (Figure \(3.2 b\) ).
- Drawing vectors with scalar magnitude \(\left|F_{A}\right|,\left|F_{B}\right|\) and a direction specified by an angle \(\phi_{A}, \phi_{B}\) (Figure 3.2c).
Although the first method of drawing without scalars can always be used, it is important to get used to the second and third method of drawing vectors as well, since they provide a way to use the drawing to provide part of the relevant information with less equations needed, and are therefore regularly used in engineering.
We note that in general, multiple 2D drawings, e.g. obtained by orthographic projection, are needed to visualise a 3D system. However, since in this textbook we will mainly deal with the dynamics in a single plane, one drawing is usually sufficient.
3.2.1 Drawing general vectors
General vectors are drawn by an arrow and a vector label like \(\overrightarrow{\boldsymbol{F}}_{A}\) (Figure 3.2a). When using this type of vector notation, the most important thing is to draw the vector with the right point of action in the drawing, to label it correctly and to make sure each relevant vector is only drawn once. Although it is convenient to draw the arrow in the direction in which the vector is expected to point, drawing it in the wrong direction will not cause problems, since both the direction and magnitude of the vector, and its conversion to scalars, are dealt with by the mathematics and a suitably chosen coordinate system. This means that in addition to the drawing an equation like \(\overrightarrow{\boldsymbol{F}}_{A}=(3 \hat{\boldsymbol{\imath}}+4 \hat{\boldsymbol{\jmath}}) \mathrm{N}\) is needed to fully specify the vector.
3.2.2 Drawing vectors with scalar signed magnitude
In some cases the direction of the line along which a vector operates is fixed, e.g. because a force operates along the line connecting two points in the drawing, but the magnitude of the vector is not, for instance because it is a time-dependent force vector or because it still needs to be calculated. In that case the vector can be represented by drawing an arrow in the direction the vector is known to act and adding a scalar label \(F_{A}\) next to it (Figure 3.2b). In this case the correct drawing of the direction of the arrow is essential for the further analysis, because it defines a unit vector \(\hat{\boldsymbol{F}}_{A}\) pointing in the same direction as the drawn arrow. Normally this unit vector is not drawn, but in this figure it is drawn to explain the concept of defining vectors with signed magnitudes. When using this method of drawing vectors, the scalar quantity \(F_{A}\) is defined as the signed magnitude of the vector \(\overrightarrow{\boldsymbol{F}}_{A}\) and the drawing defines this vector by:
\[\overrightarrow{\boldsymbol{F}}_{A} \equiv F_{A} \hat{\boldsymbol{F}}_{A} \tag{3.8} \label{3.8}\]
Note that the scalar value \(F_{A}\) can have a negative sign: that is the reason \(F_{A}\) is called a ’signed magnitude’. When \(F_{A}\) is negative, the vector \(\overrightarrow{\boldsymbol{F}}_{A}\) points in the opposite direction as the drawn arrow and unit vector \(\hat{\boldsymbol{F}}_{A}\). This also means that \(F_{A}\) can have a different sign than the vector’s magnitude which is always positive \(\left(F_{A} \pm\left|\overrightarrow{\boldsymbol{F}}_{A}\right|\right)\). If the magnitude of the vector is known and fixed, then one can replace the label \(F_{A}\) by the magnitude of the vector including the unit, for instance by writing \(3 \mathrm{~N}\) besides the arrow like in Figure 3.2b).
3.2.3 Drawing vectors by magnitude and angle
A planar \({ }^{1}\) vector \(\overrightarrow{\boldsymbol{F}}_{A}\), can also be indicated by a magnitude \(\left|\overrightarrow{\boldsymbol{F}}_{A}\right|\) and an angle \(\phi_{A}\) (Figure 3.2c). When specifying the angle \(\phi_{A}\), one also needs to specify the reference direction at which \(\phi_{A}=0\), which is indicated by a dashed line and is usually taken to be the positive \(x\)-axis. One also draws a circular arrow that points from the reference direction towards the drawn vector to indicate the direction in which the angle \(\phi_{A}\) increases. In this textbook we use the convention that in the \(x y\)-plane such an angle has a positive scalar value if it rotates in the same direction as given by the right-hand rule \({ }^{2}\). So in the coordinate system drawn in Figure 3.2 the angle increases in the counter-clockwise direction. Then, in a Cartesian coordinate system in the \(x y\)-plane (see later in this chapter) one finds that:
\[\overrightarrow{\boldsymbol{F}}_{A}=\left|\overrightarrow{\boldsymbol{F}}_{A}\right| \cos \phi_{A} \hat{\boldsymbol{\imath}} + \left|\overrightarrow{\boldsymbol{F}}_{A}\right| \sin \phi_{A} \hat{\boldsymbol{\jmath}} \tag{3.9} \label{3.9}\]
An angle between two vectors can be specified as \(\phi_{B / A}=\phi_{B}-\phi_{A}\). If the magnitude and/or angle of the vector are known and fixed, then one can replace the labels \(\left|\overrightarrow{\boldsymbol{F}}_{A}\right|\) and \(\phi_{A}\) by the magnitude and angle of the vector including the unit, for instance by writing ’ \(3 \mathrm{~N}\) ’ besides the arrow and replacing the angle \(\phi_{A}\) by \(233^{\circ}\), as can be seen in Figure 3.2c). Inverting the direction of such a vector is done by adding \(\pi \operatorname{rad}\left(=180^{\circ}\right)\) to the angle \(\left(\phi_{A} \rightarrow \phi_{A}+\pi\right)\). This is a difference with the signed magnitude vector, for which the inversion is performed by changing the sign of the signed magnitude \(\left(F_{A} \rightarrow-F_{A}\right)\).
Give the mathematical expressions for the following quantities such that the corresponding vectors in Figure 3.2 have identical magnitude and direction as the vector in Figure 3.2b) specified by label \(3 \mathrm{~N}\):
- \(\overrightarrow{\boldsymbol{F}}_{A}\) and \(\overrightarrow{\boldsymbol{F}}_{B}\) in Figure 3.2a)
- \(F_{A}\) and \(F_{B}\) in Figure 3.2b)
- The combination \(\left|\overrightarrow{\boldsymbol{F}}_{A}\right|\) and \(\phi_{A}\) in Figure 3.2c)
Exemplary Solution
- \(\overrightarrow{\boldsymbol{F}}_{A}=\frac{3}{5}(3 \hat{\imath}+4 \hat{\jmath}) \mathrm{N}\) and \(\overrightarrow{\boldsymbol{F}}_{B}=\frac{3}{5}(3 \hat{\imath}+4 \hat{\jmath}) \mathrm{N}\)
- \(F_{A}=3 \mathrm{~N}\) and \(F_{B}=-3 \mathrm{~N}\)
- \(\left|\vec{F}_{A}\right|=3 \mathrm{~N}\) and \(\phi_{A}=\arctan \frac{4}{3} \approx 53^{\circ}\)
3.2.4 Drawing fixed, sliding and free vectors
A vector only has a direction and magnitude. However, for vectors in mechanics, the position at which vectors are drawn can be essential for the correct analysis. For instance the position of the origin is essential for the interpretation of a position vector, and the point of action of a force vector can be essential for its effect on a rigid body. For this reason, vectors can be complemented by a point (reference point, or point of action), or by a line of points (line of action). The combination of the vector and this point or line is then called a fixed vector or a sliding vector.
- Fixed vectors: these are vectors which have to be drawn at a fixed point of action or reference point for the correct dynamic analysis. This point of action or reference point should be indicated by a black dot on the tail or tip of the arrow. Position vectors are fixed vectors and always have their point of reference at the tail of the arrow. The vector \(\overrightarrow{\boldsymbol{r}}_{A}\) in Figure 3.3 is an example.
- Sliding vectors: for sliding vectors, instead of a point of action, one can use a line of action, that is shown as a dashed line in Figure 3.3. The line of action of a force vector \(\overrightarrow{\boldsymbol{F}}_{B}\) is the line that passes through its point of action \(B\) and is parallel to \(\overrightarrow{\boldsymbol{F}}_{B}\). For sliding vectors any point of action that lies on the line of action can be chosen without affecting the dynamic analysis, so the vector can slide along the line. For example, to determine the moment vector of force \(\overrightarrow{\boldsymbol{F}}_{B}\) with respect to a reference point \(P\), one uses the cross product \(\overrightarrow{\boldsymbol{M}}_{B / P}=\overrightarrow{\boldsymbol{r}}_{B / P} \times \overrightarrow{\boldsymbol{F}}_{B}\). It can be shown that replacing point \(B\) by any other point on the line of action, results in the same moment vector \(\vec{M}_{B / P}\). This is visualised in Figure 3.3 by showing that the area of the red rectangle and red parallelogram are equal. When sliding a vector, the line of action should be drawn as a dashed line that passes through the point of action or reference point, which should still be indicated by a black dot, like point \(B\) in Figure 3.3. Sliding a vector along its line of action can be performed to facilitate or clarify a drawing, but is in particular useful to facilitate determining a cross product. As will be discussed in Sec. 3.4, the magnitude of such a cross product \(\overrightarrow{\boldsymbol{r}}_{B / P} \times \overrightarrow{\boldsymbol{F}}_{B}\) is equal to the area of the red parallelogram Figure 3.3. As shown by the red rectangle in Figure 3.3, one can slide the tail of the vector \(\overrightarrow{\boldsymbol{F}}_{B}\) along the line of action to the point nearest to \(P\) at a distance \(r_{\text {min }}\). Then the cross product is simply proportional to the area of the red rectangle, so \(\left|\overrightarrow{\boldsymbol{M}}_{B / P}\right|=r_{\text {min }}\left|\overrightarrow{\boldsymbol{F}}_{B}\right|\).
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Free vectors: There is no reference point for a free vector, so free vectors can be drawn anywhere. However it is often clearer to draw them
on or near the object on which they act, like shown for the vector \(\overrightarrow{\boldsymbol{M}}_{B / P}\) in Figure 3.3. The moment, couple, and angular momentum vector are free vectors. Also the angular velocity and angular acceleration vectors of a rigid body are examples of free vectors. - Vector fields: vector fields consist of vectors at every position in space. They are best drawn as vectors on the objects on which they act, while their spatial dependence is given as equations. Alternatively vector fields can be drawn and visualised at many points in space (similar to Figure 5.6). Since the gravitational acceleration vector field on earth is approximately constant, it can be drawn at a single position like shown in Figure 3.3 using signed magnitude notation \(\overrightarrow{\boldsymbol{g}}=-g \hat{\boldsymbol{\jmath}}\), by which we indicate that it is identical for all objects in the figure.
3.2.5 Projecting vectors and drawing components
After drawing the vectors and defining the coordinate system, the vectors can be projected on the coordinate axes to obtain and draw the vector components. To obtain the components along each of the axes, we take the dot product with the unit vectors. For example in Cartesian coordinates we get for the projection of the vector \(\overrightarrow{\boldsymbol{F}}_{A}\) on the coordinate system (CS) the following 3 components:
\[\begin{align} F_{A, x} & =\overrightarrow{\boldsymbol{F}}_{A} \cdot \hat{\boldsymbol{\imath}} \tag{3.10} \label{3.10}\\[4pt] F_{A, y} & =\overrightarrow{\boldsymbol{F}}_{A} \cdot \hat{\boldsymbol{\jmath}} \tag{3.11} \label{3.11}\\[4pt] F_{A, z} & =\overrightarrow{\boldsymbol{F}}_{A} \cdot \hat{\boldsymbol{k}} \tag{3.12} \label{3.12}\end{align}\]
As a result we have \(\overrightarrow{\boldsymbol{F}}_{A}=F_{A, x} \hat{\boldsymbol{\imath}}+F_{A, y} \hat{\boldsymbol{\jmath}}+F_{A, z} \hat{\boldsymbol{k}}\), and projected vector components can be considered as vectors themselves, e.g. \(\overrightarrow{\boldsymbol{F}}_{A, x}=F_{A, x} \hat{\imath}\), with axis subscripts \(x, y, z\) to indicate the projected components. Examples of drawing projected vector components are shown in Figure 3.4. Projected vector components are in practice always drawn using signed magnitude notation (so with scalar labels). If both the vector and its projected vector components are drawn, one has to use dashed arrows for the drawn vector components, to distinguish them from the original vector (see examples in Figure 3.4). This is needed to prevent errors from double counting of forces in an FBD. According to the rules for drawing vectors with signed magnitude notation, a minus sign is needed in front of the projected components when the arrow points in the negative axis direction, like for \(-F_{B, x}\) in Figure 3.4.
3.2.6 Scaling of drawn vectors
If you want to use geometric techniques to make calculations with vectors, then it is important to draw them with the correct magnitude. Euclidean vectors are drawn in a coordinate system with axes with distance dimensions and units in e.g. meters. Using these axes, the magnitude of a position vector can be determined by measuring its length. However, the magnitude of other vectors, like velocity or force (see e.g. vector \(\overrightarrow{\boldsymbol{F}}_{B}\) in Figure 3.3), that have units
of \(\mathrm{m} / \mathrm{s}\) or \(\mathrm{N}\) cannot be determined from the axes. There is therefore always a scale factor that can be chosen arbitrarily to convert the drawn length of such a vector to its actual magnitude. For example a force vector \(\overrightarrow{\boldsymbol{F}}\) that has a magnitude \(|\overrightarrow{\boldsymbol{F}}|=1 \mathrm{~N}\) can be drawn with a length of \(1 \mathrm{~m}\). In that case there is a scale factor for force vectors \(\gamma_{F}=1 \mathrm{~m} / \mathrm{N}\), but other scale factors might also be chosen. After choosing a scale factor, the drawn length of all force vectors should be determined from their magnitude using the equation \(\gamma_{F}|\overrightarrow{\boldsymbol{F}}|\). This scaling is especially important when graphical methods are used that use the drawn magnitudes of vectors.
3.2.7 Drawing points
Points are drawn as black dots with a label in a sketch or FBD. It is important to realise that there are different types of points. There are points that are fixed \({ }^{3}\) in space, but also points that move along with a rigid body, like the point \(B\) in Figure 3.3. To clarify in a drawing which points move along with a rigid body and which ones do not, we adopt the following conventions in this textbook:
- Relevant points are always indicated by a black dot.
- If the black dot is not touching a rigid body, the point is assumed to be fixed space, like the point at \((0,0)\) in Figure 3.3.
- If the black dot is touching a rigid body or its edge, the point is moving along with the rigid body (like any mass inside the rigid body), like point \(B\) and \(C\) in Figure 3.3.
- In all other cases more information needs to be provided to clarify if the point indicated by the drawn dot is fixed in space or moving along with one of the objects in the drawing. As an example see the caption of Figure 9.8, where the drawn dots touch 2 rigid bodies, and different labels are used to indicate points that move along with different rigid bodies, while being represented by the same black dot.
The main message is to be aware that there are different types of points in dynamics: points fixed in space, and points fixed to rigid bodies or other objects. Further clarification of the motion of points can be provided in a text that accompanies the drawing.
3.2.8 Drawing distances and dimensions
Often dimensions and distances (see Sec. 3.1.2) between two points needs to be drawn. Distances are always positive and drawn as a double sided arrow, with arrow heads on both of its ends and/or small perpendicular end caps. An example is shown in Figure 3.3 for the distance \(L\) that characterises the side length of the rectangle.