vector 2d add

Addition of vectors can be peformed graphically [1] or by adding the corresponding components [2]. We can also use trigonometry through law of sines and cosines.

components addition

Adding corresponding components in performing vector addition is simpler and does not require any drawing but only formula. Suppose that there is first vector

\begin{equation} \label{eqn:vadd-r1} \vec{r}_1 = x_1 \ \hat{x} + y_1 \ \hat{y} \end{equation}

and second vector

\begin{equation} \label{eqn:vadd-r2} \vec{r}_2 = x_2 \ \hat{x} + y_2 \ \hat{y} \end{equation}

that will be the operands of vector addition operation in the form of

\begin{equation} \label{eqn:vadd-addition} \vec{r}_3 = \vec{r}_1 + \vec{r}_2, \end{equation}

which gives that

\begin{equation} \label{eqn:vadd-addition-x} x_3 = x_1 + x_2 \end{equation}

and

\begin{equation} \label{eqn:vadd-addition-y} y_3 = y_1 + y_2. \end{equation}

Let’s substitute Eqns. \eqref{eqn:vadd-r1} and \eqref{eqn:vadd-r2} into Eqn. \eqref{eqn:vadd-addition} and obtain the following result

\begin{equation} \label{eqn:vadd-addition-result} \begin{array}{rcl} \vec{r}_3 & = & (x_1 \ \hat{x} + y_1 \ \hat{y}) + (x_2 \ \hat{x} + y_2 \ \hat{y}) \newline & = & x_1 \ \hat{x} + y_1 \ \hat{y} + x_2 \ \hat{x} + y_2 \ \hat{y} \newline & = & x_1 \ \hat{x} + x_2 \ \hat{x} + y_1 \ \hat{y} + y_2 \ \hat{y} \newline & = & (x_1 + x_2) \ \hat{x} + (y_1 + y_2) \ \hat{y} \newline & = & x_3 \ \hat{x} + y_3 \ \hat{y}, \end{array} \end{equation}

which simply proof Eqns. \eqref{eqn:vadd-addition-x} and \eqref{eqn:vadd-addition-y}. And how about a vector in three-dimensional space, e.g.

\begin{equation} \label{eqn:vadd-ri} \vec{r}_i = x_i \ \hat{x} + y_i \ \hat{y} + z_i \ \hat{z}, \end{equation}

with $i = 1, 2, 3$ as in Eqns. \eqref{eqn:vadd-r1} - \eqref{eqn:vadd-addition}. See Exercise 4 to for this.

graphical addition

We will discuss first vectors in two-dimensional space since they are easier to visualize than the one in three-dimensional space. The vectors are $\vec{r}_1$ and $\vec{r}_2$ as described in Eqns \eqref{eqn:vadd-r1} and \eqref{eqn:vadd-r2}. Two methods will discussed in this part.

parallelogram method

This method related to the two-dimensional form, i.e. parallelogram, that is constructed by the two vectors used as operands in the vector addition.

Fig. 1 shows that $\vec{r}_1 = 3 \ \hat{x} + \hat{y}$, $\vec{r}_2 = \hat{x} + 2 \ \hat{y}$, and $\vec{r}_3 = 4 \ \hat{x} + 3 \ \hat{y}$. The result of $\vec{r}_3$ is simply according to Eqns. \eqref{eqn:vadd-addition-x} and \eqref{eqn:vadd-addition-y}. This graphical method known as parallelogram method. You can find tutorial about this method [3]. You can see in Fig. 1 that vector $\vec{r}_1$ (solid green line) with its pair (dashed green line) and vector $\vec{r}_2$ (solid blue line) with its pair (dashed blue line) construct a parallelogram where the result is simply the diagonal between the two vectors (solid red line).

polygon method

In this method, which is simply put start point of a vector to end point of another vector [4].

Fig. 2 (right) show the difference of this method from the previous one. The start point of vector $\vec{r}_2$ is put to the end point of vector $\vec{r}_1$ and the result is from start point of the first vector to the end point of the last vector. The given example of addition of two vectors does not really show the difference between this two methods. It requires more vectors to show the advantage of the second method compared the first one.

addition of four vectors

As an example, we will considered four vectors that their sum or resultant will be calculated. Not all vector are drawn simultaneously in Fig. 3, but only one additional vector in each graph, in order showing how the method works.

Fig. 3 shows the results of $\vec{r} _{1+2} = \vec{r}_1 + \vec{r}_2$ (left), $\vec{r} _{1+2+3} = \vec{r}_3 + \vec{r} _{1+2}$ (center), and $\vec{r} _{1+2+3+4} = \vec{r}_4 + \vec{r} _{1+2+3}$ (right), where in each step only one vector is added to the other. Actually, we can also do in another way, e.g. $(\vec{r}_1 + \vec{r}_2) + (\vec{r}_3 + \vec{r}_4) = \vec{r} _{1+2} + \vec{r} _{3+4} = \vec{r} _{1+2+3+4}$.

Now let’s see how the polygon method works for those four vectors.

Fig. 4 shows the results of $\vec{r} _{1+2} = \vec{r}_1 + \vec{r}_2$ (left), $\vec{r} _{1+2+3} = \vec{r}_1 + \vec{r}_2 + \vec{r}_3$ (center), and $\vec{r} _{1+2+3+4} = \vec{r}_1 + \vec{r}_2 + \vec{r}_3 + \vec{r}_4$ (right) using polygon method. With this method all vectors can still be drawn and their contribution to the resultan are clear.

This method will be used later in summing phasors (or rotating vectors), in explaining the interference of multiple slit and diffraction.

trigonometry addition

Law of cosines and sines can also be used to calculate vector addition or resultan of two vectors [5]. Length of both vectors and the angle between them are required fot this method.

exer

1. Using similar way as in Eqn. \eqref{eqn:vadd-addition-result}, find formulation for $C_x$ and $C_y$, if $\vec{C} = \vec{A} + \vec{B}$. Use $\vec{A} = A_x \ \hat{x} + A_y \ \hat{y}$, $\vec{B} = B_x \ \hat{x} + B_y \ \hat{y}$, and $\vec{C} = C_x \ \hat{x} + C_y \ \hat{y}$.
2. Calculate $\vec{G} = \vec{D} + \vec{E}$, if $\vec{D} = 2 \ \hat{x} + 3 \ \hat{y}$ and $\vec{E} = 3 \ \hat{x} - 4 \ \hat{y}$.
3. There are two vectors, $\vec{B} = 12 \ \hat{x} + 3 \ \hat{y}$ and $\vec{C} = -7 \ \hat{x} + 9 \ \hat{y}$, calculate vector $\vec{D} = \vec{B} + \vec{C}$ and also its magnitude $|\vec{D}|$.
4. Find the formulation to calculate $x_3$, $y_3$, and $z_3$ for vector addition $\vec{r}_3 = \vec{r}_1 + \vec{r}_2$, with $\vec{r}_i$ is defined in Eqn. \eqref{eqn:vadd-ri}.
5. Fig. 3 show the process of four vectors addition $(((\vec{r}_1 + \vec{r}_2) + \vec{r}_3) + \vec{r}_4)$, draw similar graphs for the process $(\vec{r}_1 + \vec{r}_2) + (\vec{r}_3 + \vec{r}_4)$.

note

1. Carl R. Nave, “Graphical Vector Addition”, HyperPhysics, 2017, url http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html#vec1 [20200921].
2. Eric W. Weisstein, “Vector Addition”, from MathWorld–A Wolfram Web Resource, url https://mathworld.wolfram.com/VectorAddition.html [20200921].
3. Math and Stats Help, “Find the Resultant Force using the Parallelogram Method”, YouTube, 18.11.2019, url https://www.youtube.com/watch?v=WhHO77rEIfk [20200921].
4. Simon Plus, “Episode 5: Polygon Method for Vector Addition”, YouTube, 24.04.2017, url https://www.youtube.com/watch?v=r-iKebb2FIg [20200921].
5. -, “Calculating the Resultant using the Law of Cosines and Sines”, Alberta Learning, 16 Jun 2004, url http://www.learnalberta.ca/content/sep20u/html/java/vector_addition_numerical/applethelp/lesson_1.html#2 [20200922].

coord sys 2d intro • vector 2d intro