3.4: Vectors Products
To mathematically express the laws of dynamics, we will often need two types of vector products: the dot product indicated by the symbol ’, and the cross product, indicated by the symbol \(\times\). Note that these symbols only represent dot and cross products when they are placed between two vectors. If they are placed between two scalars or a scalar and a vector they indicate scalar multiplication.
3.4.1 Dot products using unit vector notation
Dot products or cross products of vectors can be evaluated using unit vector notation, using the following relations for dot products between unit vectors, that follow from their orthogonality:
\[\begin{align} & \hat{\boldsymbol{\imath}} \cdot \hat{\boldsymbol{\imath}}=\hat{\boldsymbol{\jmath}} \cdot \hat{\boldsymbol{\jmath}}=\hat{\boldsymbol{k}} \cdot \hat{\boldsymbol{k}}=1 \tag{3.23} \label{3.23}\\[4pt] & \hat{\boldsymbol{\imath}} \cdot \hat{\boldsymbol{\jmath}}=\hat{\boldsymbol{\jmath}} \cdot \hat{\boldsymbol{k}}=\hat{\boldsymbol{k}} \cdot \hat{\boldsymbol{\imath}}=0 \tag{3.24} \label{3.24}\end{align}\]
All dot products can be evaluated using these relations between the unit vectors. For example one now can take the dot product between two vectors \(\overrightarrow{\boldsymbol{v}}=v_{x} \hat{\boldsymbol{\imath}}+v_{y} \hat{\boldsymbol{\jmath}}\) and \(\overrightarrow{\boldsymbol{F}}=F_{x} \hat{\boldsymbol{\imath}}+F_{y} \hat{\boldsymbol{\jmath}}\) as follows:
\[\begin{align} \overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{F}} & =\left(v_{x} \hat{\boldsymbol{\imath}}+v_{y} \hat{\boldsymbol{\jmath}}\right) \cdot\left(F_{x} \hat{\boldsymbol{\imath}}+F_{y} \hat{\boldsymbol{\jmath}}\right) \tag{3.25} \label{3.25}\\[4pt] & =v_{x} F_{x} \hat{\boldsymbol{\imath}} \cdot \hat{\boldsymbol{\imath}}+v_{y} F_{x} \hat{\boldsymbol{\jmath}} \cdot \hat{\boldsymbol{\imath}}+v_{x} F_{y} \hat{\imath} \cdot \hat{\boldsymbol{\jmath}}+v_{y} F_{y} \hat{\boldsymbol{\jmath}} \cdot \hat{\boldsymbol{\jmath}} \tag{3.26} \label{3.26}\\[4pt] & =v_{x} F_{x} \times 1+v_{y} F_{x} \times 0+v_{x} F_{y} \times 0+v_{y} F_{y} \times 1 \tag{3.27} \label{3.27}\\[4pt] & =v_{x} F_{x}+v_{y} F_{y} \tag{3.28} \label{3.28}\end{align}\]
So you just multiply the scalar values by the dot products of the unit vectors. Here we used Equation \ref{3.23} and Equation \ref{3.24} to determine the dot products between the unit vectors like \(\hat{\boldsymbol{\imath}} \cdot \hat{\boldsymbol{\imath}}\).
3.4.2 Cross product using unit vector notation
For cross products the relations between unit vectors in a right-handed axes system are as follows:
\[\begin{align} \hat{\boldsymbol{\imath}} \times \hat{\boldsymbol{\imath}}=\hat{\boldsymbol{\jmath}} \times \hat{\boldsymbol{\jmath}}=\hat{\boldsymbol{k}} \times \hat{\boldsymbol{k}} & =\overrightarrow{\mathbf{0}} \tag{3.29} \label{3.29}\\[4pt] \hat{\boldsymbol{\imath}} \times \hat{\boldsymbol{\jmath}}=-\hat{\boldsymbol{\jmath}} \times \hat{\boldsymbol{\imath}} & =\hat{\boldsymbol{k}} \tag{3.30} \label{3.30}\\[4pt] \hat{\boldsymbol{\jmath}} \times \hat{\boldsymbol{k}}=-\hat{\boldsymbol{k}} \times \hat{\boldsymbol{\jmath}} & =\hat{\boldsymbol{\imath}} \tag{3.31} \label{3.31}\\[4pt] \hat{\boldsymbol{k}} \times \hat{\boldsymbol{\imath}}=-\hat{\boldsymbol{\imath}} \times \hat{\boldsymbol{k}} & =\hat{\boldsymbol{\jmath}} \tag{3.32} \label{3.32}\end{align}\]
Where we used that for every cross product it holds that \(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}=-\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{A}}\) (whereas for dot products \(\overrightarrow{\boldsymbol{A}} \cdot \overrightarrow{\boldsymbol{B}}=\overrightarrow{\boldsymbol{B}} \cdot \overrightarrow{\boldsymbol{A}}\) ). With these relations all cross products can be evaluated using the unit vectors. For example one now can take the cross product between two vectors \(\overrightarrow{\boldsymbol{r}}=r_{x} \hat{\boldsymbol{\imath}}+r_{y} \hat{\boldsymbol{\jmath}}\) and \(\overrightarrow{\boldsymbol{F}}=F_{x} \hat{\boldsymbol{\imath}}+F_{y} \hat{\boldsymbol{\jmath}}\) as follows:
\[\begin{align} \overrightarrow{\boldsymbol{r}} \times \overrightarrow{\boldsymbol{F}} & =\left(r_{x} \hat{\boldsymbol{\imath}}+r_{y} \hat{\boldsymbol{\jmath}}\right) \times\left(F_{x} \hat{\boldsymbol{\imath}}+F_{y} \hat{\boldsymbol{\jmath}}\right) \\[4pt] & =r_{x} F_{x} \hat{\boldsymbol{\imath}} \times \hat{\boldsymbol{\imath}}+r_{y} F_{x} \hat{\boldsymbol{\jmath}} \times \hat{\boldsymbol{\imath}}+r_{x} F_{y} \hat{\imath} \times \hat{\boldsymbol{\jmath}}+r_{y} F_{y} \hat{\boldsymbol{\jmath}} \times \hat{\boldsymbol{\jmath}} \\[4pt] & =r_{x} F_{x} \cdot \overrightarrow{\mathbf{0}}+r_{y} F_{x} \cdot(-\hat{\boldsymbol{k}})+r_{x} F_{y} \cdot(+\hat{\boldsymbol{k}})+r_{y} F_{y} \cdot \overrightarrow{\mathbf{0}} \\[4pt] & =\left(r_{x} F_{y}-r_{y} F_{x}\right) \hat{\boldsymbol{k}} \tag{3.33} \label{3.33}\end{align}\]
Where we used Equation \ref{3.29} and Equation \ref{3.30} to evaluate the cross products between the unit vectors. Using unit vector notation has several advantages. You only have to write down and evaluate those unit vectors that are non-zero, the notation takes much less space and it is immediately clear what type of coordinate system one is working in from the unit vectors. For these reasons we will usually work with unit vector notation in this textbook.
3.4.3 Graphical analysis of dot and cross products
Dot and cross products can also be evaluated graphically as is shown in Figure 3.8. The two vectors are drawn such that their tails are at the same point in space and the planar angle \(\phi_{B / C}\) between them is determined. To obtain the dot product one projects one of the vectors on the direction of the other, obtaining
the scalar \(\left|F_{B}\right| \cos \phi_{B / C}\), which is then multiplied by the magnitude of the other vector \(\left|v_{C}\right|\) giving the result:
\[\overrightarrow{\boldsymbol{v}}_{C} \cdot \overrightarrow{\boldsymbol{F}}_{B}=\left|\overrightarrow{\boldsymbol{v}}_{C}\right|\left|\overrightarrow{\boldsymbol{F}}_{B}\right| \cos \phi_{B / C} \tag{3.34} \label{3.34}\]
The magnitude of this scalar value is proportional to the area of the pink parallelogram shown in Figure 3.8. Note that the direction of the distance \(\left|v_{C}\right|\) should be drawn at \(90^{\circ}\) to the actual vector \(\overrightarrow{\boldsymbol{v}}_{C}\) to construct this parallelogram for the dot product.
For graphically analysing the cross product of two vectors \(\overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B}\) (see Figure 3.8), one puts their tails together and draws a circular angle arrow \(\phi_{B / A}\), that starts at the first vector \(\overrightarrow{\boldsymbol{r}}_{A}\) (left of the \(\times\) ) and points anticlockwise towards the second vector \(\overrightarrow{\boldsymbol{F}}_{B}\). Then if both vectors are in the \(x y\)-plane, their cross product can be determined as:
\[\overrightarrow{\boldsymbol{M}}_{B / A}=\overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B}=\left|\overrightarrow{\boldsymbol{r}}_{A}\right|\left|\overrightarrow{\boldsymbol{F}}_{B}\right| \sin \phi_{B / A} \hat{\boldsymbol{k}} \tag{3.35} \label{3.35}\]
As can be seen in Figure 3.8, the magnitude of the cross product vector is equal to the red parallelogram whose sides are formed by the two vectors. The angle \(\phi_{B / A}\) is measured in the anticlockwise direction. Since \(\phi_{B / A}>180^{\circ}\), we have \(\sin \phi_{B / A}<0\) and the vector \(\overrightarrow{\boldsymbol{M}}_{B / A}\) points in the \(-\hat{\boldsymbol{k}}\) direction, into the plane as indicated by the \(\otimes\) sign.
Right-hand rule
The direction of the cross-product vector \(\overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B}\) can also conveniently be determined using the right-hand rule. In that case it is important not to always draw an anticlockwise angle, but to draw the smallest angle \(\phi_{B / A}^{\prime}\) from the first vector \(\overrightarrow{\boldsymbol{r}}_{A}\) in the cross product to the second vector \(\overrightarrow{\boldsymbol{F}}_{B}\) (see dashed arrow \(\phi_{B / A}^{\prime}\) in Figure 3.8). So, make sure \(\phi_{B / A}^{\prime}<180^{\circ}\). Then curve the fingers of your right hand parallel to that curved arrow, with fingers pointing in the same direction as the arrowhead. Your thumb will then point in the same direction as \(\overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B}\) in the \(-\hat{\boldsymbol{k}}\) direction. It is recommended to always check the direction of vectors resulting from cross-product calculations using the right-hand rule. There are several alternative variations of right-hand and left-hand rules to determine the direction of a cross-product. Feel free to choose and memorise the one that you find most easy to use.
- Determine the values of the dot product \(\overrightarrow{\boldsymbol{v}}_{C} \cdot \overrightarrow{\boldsymbol{F}}_{B}\) and the cross product \(\overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B}\) using unit vector notation. Use that the length of the vectors in Figure 3.8 is scaled such that \(1 \mathrm{~m}: 1 \mathrm{~m} / \mathrm{s}: 1 \mathrm{~N}\).
- Determine the values of the dot product \(\overrightarrow{\boldsymbol{v}}_{C} \cdot \overrightarrow{\boldsymbol{F}}_{B}\) and the cross product \(\overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B}\) using graphical analysis.
Solution
Part a
Since all vectors are in the \(x y\)-plane we can reuse Equation \ref{3.28} and use Figure 3.8 to obtain:
\[\begin{align} \overrightarrow{\boldsymbol{v}}_{C} \cdot \overrightarrow{\boldsymbol{F}}_{B} & =v_{C, x} F_{B, x}+v_{C, y} F_{B, y} \tag{3.36} \label{3.36}\\[4pt] & =[(-3 \times-2)+(0 \times-4)] \mathrm{N} \cdot \mathrm{m} / \mathrm{s} \tag{3.37} \label{3.37}\\[4pt] & =6 \mathrm{~N} \cdot \mathrm{m} / \mathrm{s} \tag{3.38} \label{3.38}\end{align}\]
For the cross product between two vectors in the \(x y\)-plane we can use the result from Equation \ref{3.33}:
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B} & =\left(r_{A, x} F_{B, y}-r_{A, y} F_{B, x}\right) \hat{\boldsymbol{k}} \tag{3.39} \label{3.39}\\[4pt] & =(6 \times(-4)-2 \times(-2)) \mathrm{N} \cdot \mathrm{m} \hat{\boldsymbol{k}} \tag{3.40} \label{3.40}\\[4pt] & =-20 \mathrm{~N} \cdot \mathrm{m} \hat{\boldsymbol{k}} \tag{3.41} \label{3.41}\end{align}\]
Part b
First we determine the angles of the vectors with respect to the positive \(x\)-axis direction. We have \(\phi_{A}=\arctan \frac{2}{6}=18.4^{\circ}, \phi_{B}=180^{\circ}+\arctan \frac{-4}{-2}=243.4^{\circ}\) and \(\phi_{C}=\arctan \frac{0}{-3}=180^{\circ}\). Then we find \(\phi_{B / C}=\phi_{B}-\phi_{C}=63.4^{\circ}\) and \(\phi_{B / A}=\phi_{B}-\) \(\phi_{A}=225^{\circ}\).
We determine the magnitude of the three vectors:
\[\left|\overrightarrow{\boldsymbol{r}}_{A}\right|=\sqrt{r_{A, x}^{2}+r_{A, y}^{2}}=\sqrt{40} \mathrm{~m} \nonumber\]
\[\left|\overrightarrow{\boldsymbol{F}}_{B}\right|=\sqrt{F_{B, x}^{2}+F_{B, y}^{2}}=\sqrt{20} \mathrm{~N} \nonumber\]
\[\left|\overrightarrow{\boldsymbol{v}}_{C}\right|=\sqrt{v_{C, x}^{2}+v_{C, y}^{2}}=3 \mathrm{~m} / \mathrm{s} \nonumber\]
Filling these numbers in Equation \ref{3.34} and Equation \ref{3.35} we find:
\[\begin{align} \overrightarrow{\boldsymbol{v}}_{C} \cdot \overrightarrow{\boldsymbol{F}}_{B} & =3 \mathrm{~m} / \mathrm{s} \times \sqrt{20} \mathrm{~N} \cos \left(63.4^{\circ}\right) \tag{3.42} \label{3.42}\\[4pt] & =6 \mathrm{~N} \cdot \mathrm{m} / \mathrm{s} \tag{3.43} \label{3.43}\\[4pt] \overrightarrow{\boldsymbol{r}}_{A} \times \overrightarrow{\boldsymbol{F}}_{B} & =\sqrt{40} \mathrm{~m} \times \sqrt{20} \mathrm{~N} \sin \left(225^{\circ}\right) \hat{\boldsymbol{k}} \tag{3.44} \label{3.44}\\[4pt] & =-20 \mathrm{~N} \cdot \mathrm{m} \hat{\boldsymbol{k}} \tag{3.45} \label{3.45}\end{align}\]
So, as it should be, both the unit vector and graphical method to determine dot and cross-products give the same result.