Lesson Explainer: Vector Addition Physics

In this explainer, we will learn how to add two or more vectors in two dimensions, using both graphical and algebraic methods.

Recall that a vector is a quantity that has both a magnitude and a direction.

The grid below shows two vectors, represented by arrows.

The magnitude of each vector is represented by the length of each arrow. The two arrows shown in the diagram have the same length of four grid squares, so the two vectors have the same magnitude. However, they point in different directions. The blue vector points along the π‘₯-axis, while the red vector points along the 𝑦-axis.

The grid below shows two different vectors.

Both the green vector and the orange vector point in the same direction, but they have different lengths. The orange vector has a length of 3 grid squares, while the green vector has a length of 6 grid squares.

In this explainer, the symbol for a vector will be written with a half-arrow on top of it, for example, ⃑𝐴. In other media, you may see vectors notated in different ways, for example, in a bold font like A.

Now look at the two vectors shown on the grid below.

What would the result of adding vectors ⃑𝐴 and ⃑𝐡 together be? We can work this out just using the diagram. Imagine moving vector ⃑𝐡 so that the β€œtail” of the arrow (the end without the arrowhead) is at the same point on the grid as the β€œtip” of the arrow (the end with the arrowhead) that represents vector ⃑𝐴. This is shown in the diagram below.

Notice that the length and direction of vector ⃑𝐡 have not changed. It has simply been translated across the grid. The sum of the two vectors is now given by a vector, ⃑𝑉, that goes from the β€œtail” of vector ⃑𝐴 to the β€œtip” of vector ⃑𝐡, as shown by the magenta arrow in the diagram below.

We could also have done this the other way around. We could have moved the tail of ⃑𝐴 to the tip of vector ⃑𝐡, and we would have gotten the same result, as shown below.

When adding vectors using this method, it does not matter which order we add them in, as long as we place the vectors tip-to-tail, without changing the length or direction of each vector.

We can also use this method to add more than two vectors together. The diagram below shows three vectors on a grid.

We can find the sum of the three vectors, ⃑𝑉, by placing them tip-to-tail, as shown below.

The resultant vector, ⃑𝑉, always goes from the tail of the first vector to the tip of the last vector. This method can be used to add any number of vectors.

Let’s have a look at some examples.

Example 1: Adding Two Vectors Graphically

Which of the vectors ⃑𝑃, ⃑𝑄, ⃑𝑅, ⃑𝑆, and ⃑𝑇 shown in the diagram is equal to ⃑𝐴+⃑𝐡?

Answer

Let’s start by redrawing the diagram, highlighting vectors ⃑𝐴 and ⃑𝐡 and graying out the rest.

We can find the sum of ⃑𝐴 and ⃑𝐡 graphically by moving vector ⃑𝐡 so that the β€œtail” of the arrow is on the β€œtip” of the arrow that represents vector ⃑𝐴. This is shown below.

The resultant vector is the one that goes from the tail of vector ⃑𝐴 to the tip of vector ⃑𝐡, which is vector ⃑𝑄.

Example 2: Adding Three Vectors Graphically

Which of the vectors ⃑𝑃, ⃑𝑄, ⃑𝑅, ⃑𝑆, and ⃑𝑇 shown in the diagram is equal to ⃑𝐴+⃑𝐡+⃑𝐢?

Answer

Let’s start by redrawing the diagram, highlighting vectors ⃑𝐴, ⃑𝐡, and ⃑𝐢 and graying out the rest.

We can find the sum of ⃑𝐴, ⃑𝐡, and ⃑𝐢 graphically by moving vectors ⃑𝐡 and ⃑𝐢 so that the β€œtails” of each arrow are on the β€œtips” of each preceding arrow. This is shown below.

The resultant vector is the one that goes from the tail of vector ⃑𝐴 to the tip of vector ⃑𝐢, which is vector ⃑𝑄.

Recall that we can also represent vectors algebraically. In the diagram below, vector ⃑𝐴 can be written as 2⃑𝑖+3⃑𝑗, where ⃑𝑖 and ⃑𝑗 are unit vectors. A unit vector is a vector with a length of 1 that points along one of the axes. The unit vector ⃑𝑖 points along the π‘₯-axis, and the unit vector ⃑𝑗 points along the 𝑦-axis. The horizontal component of ⃑𝐴 has a length of 2 grid squares, so its horizontal component can be described as 2⃑𝑖, or β€œtwo times the unit vector along the π‘₯-axis.” The vertical component of ⃑𝐴 has a length of 3 grid squares, so its vertical component can be described as 3⃑𝑗, or β€œthree times the unit vector along the 𝑦-axis.” Thus, vector ⃑𝐴=2⃑𝑖+3⃑𝑗.

If we know what the horizontal and vertical components of two or more vectors are, we can find the sum of those vectors algebraically.

The diagram below shows two vectors.

We can see from the diagram that vector ⃑𝐴 has a length of 4 grid squares in the π‘₯-direction, and 1 grid square in the 𝑦-direction. Vector ⃑𝐡 has a length of 3 grid squares in the π‘₯-direction, and 3 grid squares in the 𝑦-direction. We can write this as ⃑𝐴=4⃑𝑖+1⃑𝑗,⃑𝐡=3⃑𝑖+3⃑𝑗.

In order to find what ⃑𝐴+⃑𝐡 is, we simply add each of the π‘₯-components and each of the 𝑦-components, which gives us ⃑𝐴+⃑𝐡=(4+3)⃑𝑖+(1+3)⃑𝑗⃑𝐴+⃑𝐡=7⃑𝑖+4⃑𝑗.

Note that if one of the components of a vector has a minus sign in front of it, we must include this sign when we add the π‘₯- and 𝑦-components of two or more vectors together. For example, if ⃑𝐴=4βƒ‘π‘–βˆ’2⃑𝑗, we should think of this as being ⃑𝐴=4⃑𝑖+(βˆ’2)⃑𝑗. So if we were to add the two vectors ⃑𝐴=4βƒ‘π‘–βˆ’2⃑𝑗,⃑𝐡=3⃑𝑖+3⃑𝑗, for the 𝑦-components, we would add βˆ’2 and 3, and we would get ⃑𝐴+⃑𝐡=(4+3)⃑𝑖+((βˆ’2)+3)⃑𝑗⃑𝐴+⃑𝐡=7⃑𝑖+1⃑𝑗.

Let’s have a look at some more example questions.

Example 3: Adding Two Vectors Given in Component Form

Consider two vectors ⃑𝐴 and ⃑𝐡, where ⃑𝐴=2⃑𝑖+3⃑𝑗 and ⃑𝐡=7⃑𝑖+5⃑𝑗. Calculate ⃑𝐴+⃑𝐡.

Answer

In order to find ⃑𝐴+⃑𝐡, we must add the π‘₯-components of each vector together and the 𝑦-components of each vector together, so ⃑𝐴+⃑𝐡=(2+7)⃑𝑖+(3+5)⃑𝑗⃑𝐴+⃑𝐡=9⃑𝑖+8⃑𝑗.

We now have the sum of these two vectors, written in component form.

Example 4: Adding Two Vectors Given in Component Form

Consider two vectors: ⃑𝐴 and ⃑𝐡. ⃑𝐴=3βƒ‘π‘–βˆ’3⃑𝑗 and ⃑𝐡=βˆ’4⃑𝑖+9⃑𝑗. Calculate ⃑𝐴+⃑𝐡.

Answer

In order to find ⃑𝐴+⃑𝐡, we must add the π‘₯-components of each vector together and the 𝑦-components of each vector together. We must not forget to include the minus signs before the numbers in our calculation. Doing this, we get ⃑𝐴+⃑𝐡=(3+(βˆ’4))⃑𝑖+((βˆ’3)+9)⃑𝑗⃑𝐴+⃑𝐡=βˆ’1⃑𝑖+6⃑𝑗.

We now have the sum of these two vectors, written in component form.

We can also relate adding two vectors graphically to adding two vectors algebraically, as is the case in the next example question.

Example 5: Adding Two Vectors Shown on a Diagram and Giving the Result in Component Form

The diagram shows two vectors: ⃑𝐴 and ⃑𝐡. The grid squares in the diagram have a side length of 1. What is ⃑𝐴+⃑𝐡 in component form?

Answer

There are two ways we can approach solving this problem.

The first way is that we could add the vectors graphically and then find the components of the result. The diagram below shows the two vectors added together, where vector ⃑𝐡 is moved so that its tail is on the tip of vector ⃑𝐴. Vector ⃑𝑉 is the result.

We can see from the diagram that vector ⃑𝑉 has a horizontal component of βˆ’3⃑𝑖 and a vertical component of 5⃑𝑗, so it can be written as ⃑𝑉=βˆ’3⃑𝑖+5⃑𝑗. This is our answer.

The second way we could have worked this out is simply by finding the vector components of ⃑𝐴 and ⃑𝐡 and then adding the π‘₯-components of each vector together and the 𝑦-components of each vector together. From the original diagram, we can see that ⃑𝐴=2⃑𝑖+4⃑𝑗,⃑𝐡=βˆ’5⃑𝑖+1⃑𝑗, so ⃑𝐴+⃑𝐡=(2+(βˆ’5))⃑𝑖+(4+1)⃑𝑗⃑𝐴+⃑𝐡=βˆ’3⃑𝑖+5⃑𝑗.

As you can see, we get the same result. Whether we add two vectors graphically or algebraically, we are performing the same operation on the vectors.

Key Points

  • We can add two or more vectors graphically by placing each of the vectors so that the β€œtail” of each vector is located at the β€œtip” of the preceding vector.
  • We can add two or more vectors algebraically by adding the π‘₯-components of each vector together and the 𝑦-components of each vector together.
  • Adding vectors graphically and algebraically are two different ways of performing the same operation on vectors.

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