How do you set up vectors for graphical addition using the Triangle Rule?

Does it matter which vector you start with when using the Triangle Rule?

Why can you separate a two-dimensional vector equation into two independent equations to solve for up to two unknowns?

If you and another student define vectors using different direction coordinate systems, will you end up with the same resultant vector?

What is the preferred technique to add vectors in three-dimensional systems?

In this section we will look at several different methods of vector addition. Vectors being added together are called the components, and the sum of the components is called the resultant vector.

These different methods are tools for your statics toolbox. They give you multiple different ways to think about vector addition and to approach a problem. Your goal should be to learn to use them all and to identify which approach will be the easiest to use in a given situation.

Subsection2.6.1Triangle Rule of Vector Addition

All methods of vector addition are ultimately based on the tip-to-tail method discussed in a one-dimensional context in Subsection 2.2.1. There are two ways to draw or visualize adding vectors in two or three dimensions, the Triangle Rule and Parallelogram Rule. Both are equivalent.

Triangle Rule.

Place the tail of one vector at the tip of the other vector, then draw the resultant from the first vector’s tail to the final vector’s tip.

Parallelogram Rule.

Place both vectors’ tails at the origin, then complete a parallelogram with lines parallel to each vector through the tip of the other. The resultant is equal to the diagonal from the tails to the opposite corner.

The interactive below shows two forces \(\vec{A}\) and \(\vec{B}\) pulling on a particle at the origin, and the appropriate diagram for the triangle or parallelogram rule. Both approaches produce the same resultant force \(\vec{R}\) as expected.

Subsection2.6.2Orthogonal Components

Any arbitrary vector \(\vec{F}\) can broken into two component vectors which are the sides of a parallelogram having \(\vec{F}\) as its diagonal. The process of finding components of a vector in particular directions is called vector resolution.

While a vector can be resolved into components in any two directions, it’s generally most useful to resolve them into rectangular or orthogonal components, where the parallelogram is a rectangle and the sides are perpendicular.

One benefit of finding orthogonal components is that each component is independent of the other. This independence simplifies the vector computations by allowing us to use independent equations for each orthogonal direction. Another benefit of components parallel to the coordinate axes is that you can treat these components as scalar quantities and use ordinary algebra to work with them.

However, there are an infinite number of possible rectangles to choose from, so each vector has an infinite number of sets of rectangular components. Of these, the most important one is found when the sides of the rectangle are parallel to the \(x\) and \(y\) axes. These particular components are given \(x\) and \(y\) subscripts indicate that the components are aligned with the \(x\) and \(y\) axes. For the vector \(\vec{F}\text{,}\)

where \(F_x\) and \(F_y\) are the scalar components of \(\vec{F}\text{.}\)

Another possibility is to rotate the coordinate system to any other convenient angle, and find the components in the directions of the rotated coordinate axes \(x'\) and \(y'\text{.}\) In either case, the vector is the sum of the rectangular components

The interactive below can help you visualize the relationship between a vector and its components in both the \(x\)-\(y\) and \(x'\)-\(y'\) directions.

Subsection2.6.3Graphical Vector Addition

Graphical vector addition involves drawing a scaled diagram using either the parallelogram or triangle rule, and then measuring the magnitudes and directions from the diagram. Graphical solutions work well enough for two-dimensional problems where all the vectors live in the same plane and can be drawn on a sheet of paper, but are not very useful for three-dimensional problems unless you use technology.

If you carefully draw the triangle accurately to scale and use a protractor and ruler you can measure the magnitude and direction of the resultant. However, your answer will only be as precise as your diagram and your ability to read your tools. If you use technology such as GeoGebra or a CAD program to make the diagram, your answer will be precise. The interactive in Figure 2.6.1 may be useful for this.

Even though the graphical approach has limitations, it is worth your attention because it provides a good way to visualize the effects of multiple forces, to quickly estimate ballpark answers, and to visualize the diagrams you need to use alternate methods to follow.

Subsection2.6.4Trigonometric Vector Addition

You can get a precise answer from the triangle or parallelogram rule by

drawing a quick diagram using either rule,

identifying three known sides or angles,

using trigonometry to solve for the unknown sides and angles.

The trigonometric tools you will need is in Appendix B.

Using triangle-based geometry to solve vector problems is a quick and powerful tool, but includes the following limitations:

There are only three sides in a triangle; thus vectors can only be added two at a time. If you need to add three or more vectors using this method, you must add the first two, then add the third to that sum and so on.

If you fail to draw the correct vector triangle or identify the known sides and angles, you will not find the correct answer.

The trigonometric functions are scalar functions. They are quick ways of solving for the magnitudes of vectors and the angle between vectors, but you may still need to find the vector components from a given datum.

When you need to find the resultant of more than two vectors, it is generally best to use the algebraic methods described below.

Subsection2.6.5Algebraic Addition of Components

While the parallelogram rule and the graphical and trigonometric methods are useful tools for visualizing and finding the sum of two vectors, they are not particularly suited for adding more than two vectors or working in three dimensions.

Consider vector \(\vec{R}\) which is the sum of several vectors \(\vec{A}\text{,}\)\(\vec{B}\text{,}\)\(\vec{C}\) and perhaps more. Vectors \(\vec{A}\text{,}\)\(\vec{B}\) and \(\vec{C}\) are the components of \(\vec{R}\text{,}\) and the \(\vec{R}\) is the resultant of \(\vec{A}\text{,}\)\(\vec{B}\) and \(\vec{C}\text{.}\)

It is easy enough to say that \(\vec{R} = \vec{A}+ \vec{B}+ \vec{C}\text{,}\) but how can we calculate \(\vec{R}\) if we know the components? You could draw the vectors arranged tip-to-tail and then use the triangle rule to add the first two components, then use it again to add the third component to that sum, and so forth until all the components have been added. The final sum is the resultant, \(\vec{R}\text{.}\) The process gets progressively more tedious the more components there are to sum.

This section introduces an alternate method to add multiple vectors which is straightforward, efficient and robust. This is called algebraic method, because the vector addition is replaced with a process of scalar addition of scalar components. The algebraic technique works equally well for two and three-dimensional vectors, and for summing any number of vectors.

To find the sum of multiple vectors using the algebraic:

Find the scalar components of each component vector in the \(x\) and \(y\) directions using the P to R procedure described in Subsection 2.3.3.

Algebraically sum the scalar components in each coordinate direction. The scalar components will be positive if they point right or up, negative if they point left or down. These sums are the scalar components of the resultant.

Resolve the resultant’s components to find the magnitude and direction of the resultant vector using the R to P procedure described in Subsection 2.3.3.

We can write the equation for the resultant \(\vec{F}_R\) as

This process is illustrated in the following interactive diagram and in the next example.

Example2.6.4.Vector Addition.

Vector \(\vec{A} = \N{200} \angle \ \ang{45}\) counter-clockwise from the \(x\) axis, and vector \(\vec{B} = \N{300}\)\(\angle \ang{70}\) counter-clockwise from the \(y\) axis.

Find the resultant \(\vec{R} = \vec{A} + \vec{B}\) by addition of scalar components.

Answer.

\begin{equation*}
\vec{R} = \N{281.6} \angle \ang{119.9} \textrm{ counter-clockwise from the } x \textrm{ axis}\text{.}
\end{equation*}

Solution.

Use the given information to draw a sketch of the situation. By imagining or sketching the parallelogram rule, it should be apparent that the resultant vector points up and to the left.

This answer indicates that the resultant points down and to the left. This is because the calculator answers for the inverse trig function will always be in the first or fourth quadrant. To get the actual direction of the resultant, add \(\ang{180}\) to the calculator result.

The process for adding vectors in space is exactly the same as in two dimensions, except that an additional \(z\) component is included. This interactive allows you to input the three-dimensional vector components of forces \(\vec{A}\) and \(\vec{B}\) and view the resultant force \(\vec{R}\) which is the sum of \(\vec{A}\) and \(\vec{B}\text{.}\)

Subsection2.6.6Vector Subtraction

Like one-dimensional vector subtraction, the easiest way to handle two-dimensional vector subtraction is by taking the negative of a vector followed by vector addition. Multiplying a vector by -1 preserves its magnitude but flips its direction, which has the effect of changing the sign of the scalar components.