In this explainer, we will learn how to define vectors, write vectors, and identify the geometric interpretation of basic vector operations.
We start by recalling that a scalar is a mathematical object that has size but no direction; for example, we might have a length of 4 m or a speed of 5 m/s. To represent quantities that have direction as well as size, such as a displacement of 4 m east or a velocity of 5 m/s upward, we need a different concept: the vector.
Note that throughout this explainer we will be working with vectors in the 2D plane.
Definition: Vector
A vector is a mathematical object that has both magnitude (size) and direction.
Geometrically, we can represent a vector as a directed line segment, where the length of the line segment denotes the magnitude and the orientation of the arrow shows the direction.
Some vectors are defined in terms of their initial and terminal points; so, the first vector above is .
For other vectors, we use bold lowercase letters, like above; this is usually handwritten as .
We also meet vectors that combine both sorts of labeling, such as above.
The actual location of a vector within the 2D plane is unimportant because a vector represents a translation rather than an object with a fixed position. So, for example, the three vectors labeled below are all regarded as equivalent.
If two vectors are equivalent, they must be parallel and have the same magnitudes and directions. Conversely, if two vectors are parallel and have the same magnitudes and directions, they must be equivalent.
For any vector , the vector with the same magnitude but opposite direction, which, therefore, has its arrow pointing in the opposite direction, is .
In general, for any nonzero scalar , the vector is a vector with times the magnitude of . If , the vector has the same direction as ; if , the vector has the opposite direction to . A few examples are shown below for the cases , 2, and .
Two vectors are classed as parallel if they have the same direction or are in exactly opposite directions. Therefore, all of the above vectors are parallel. Parallel vectors can always be written as nonzero scalar multiples of each other.
Next, we consider how to combine vectors.
Law: The Triangle Law for Vector Addition
We add two vectors together by positioning the head of the first vector at the tail of the second vector as shown in the left-hand diagram below.
A resultant vector is defined as a single vector whose effect is the same as the combined effects of two (or more) vectors. The resultant of two vectors starts from the tail of the first vector and finishes at the tip of the second one, forming the third side of a triangle. It represents the sum of the two vectors.
In this case, as shown in the right-hand diagram, the resultant vector goes from the tail of vector to the tip of vector , forming a triangle. It represents the sum .
This construction demonstrates the triangle law for vector addition, which states that when two vectors are represented in magnitude and direction as two sides of a triangle taken in order, the third side of the triangle, directed oppositely, represents the magnitude and direction of their sum.
By using this method, we can add together as many vectors as we like by repeatedly applying the triangle law. The diagram below demonstrates how we can extend the above process to add three vectors together.
Let us now try an example involving vector addition.
Example 1: Understanding Vector Addition Graphically
is a parallelogram with and .
Find .
Answer
Recall that a parallelogram has two pairs of parallel sides, with opposite sides having equal lengths. Also, recall that a vector is a mathematical object that has both magnitude and direction and is not defined by its location.
In parallelogram , vector is represented as the directed line segment . As side of the parallelogram is opposite to side , these sides must be parallel and equal in length. Therefore, the corresponding vectors must be parallel and equal in magnitude.
Similarly, vector is represented as the directed line segment . As side of the parallelogram is opposite to side , these sides must be parallel and equal in length. Again, the corresponding vectors must be parallel and equal in magnitude.
Recall that if two vectors are parallel and equal in magnitude, they must be equivalent. Consequently, the above observations imply that and This means that we can fill in two more vectors on the diagram as shown below.
Now recall the triangle law for vector addition, which states that when two vectors are represented in magnitude and direction as two sides of a triangle taken in order, the third side of the triangle, directed oppositely, represents the magnitude and direction of their sum.
To find , note that it is the resultant vector of and in . Therefore, by applying the triangle law, we have Equivalently, in , the vector is the resultant vector of and . This time, applying the triangle law, we have As expected, this gives the same result.
In answering the above question, we have, in fact, shown how to prove the parallelogram law of vector addition. This law states that if two vectors are represented by two adjacent sides of a parallelogram, directed away from the common vertex, with magnitudes proportional to the corresponding lengths of the sides, then their resultant vector is represented by the diagonal passing through the common vertex and directed away from it.
Next, we investigate how to subtract vectors. As we will demonstrate, it turns out that we can still apply the triangle law for vector addition in this context. For instance, suppose we have vectors and , as shown below, and we wish to work out .
First, we reverse the direction of to get . Then, positioning the head of vector at the tail of vector , we can now treat this as an addition. Applying the triangle law for vector addition, we get the resultant vector .
Note that a special case of this rule occurs when subtracting from . Positioning the head of the first vector at the tail of the second vector means that we reverse our direction and travel back to where we started once we reach the end of vector . Hence, in this case, the resultant vector is , the zero vector. Another way to think of this operation is that if , then ; so, we go from to and then back from to , arriving back where we started with zero displacement.
Let us now try an example involving vector subtraction.
Example 2: Understanding Vector Subtraction Graphically
In the diagram below, , , , and .
Write the following vectors in terms of , , , and :
Answer
Recall that a vector is a mathematical object that has both magnitude and direction.
I. We start by finding . Note that the diagram features the vector , which goes from to . Therefore, to go from to , we simply reverse the direction of this vector to get its negative, so .
II. Next, we must find . First, recall the triangle law for vector addition, which states that when two vectors are represented in magnitude and direction as two sides of a triangle taken in order, the third side of the triangle, directed oppositely, represents the magnitude and direction of their sum.
Notice that in , vector would be the resultant of and . We are given the vector , which goes from to . To go from to , we reverse the direction of this vector to get its negative, so . Moreover, we are also told that . Therefore, we can apply the triangle law to get
III. To find notice that in , vector would be the resultant of and . We have from part I, and the question tells us that . Applying the triangle law, we have
IV. Lastly, we must find . This time, notice that if we were to form the triangle , as shown below, then vector would be the resultant of and .
We already know that , and in part III we found that . Therefore, applying the triangle law, we have
In our next example, we investigate parallel vectors.
Example 3: Finding an Unknown in a Vector given a Parallel Vector
Suppose and are a pair of nonparallel vectors and that the vectors and are parallel. Work out the value of .
Answer
Recall that parallel vectors can always be written as nonzero scalar multiples of each other. Hence, in this case, we can write for some nonzero scalar . Expanding the brackets on the right-hand side, we get the equation We start by working out the value of . To do this, we equate the coefficients of on either side of the above equation, which gives
Dividing both sides by 4 and simplifying the resultant fraction, we get .
To find the value of , we equate the coefficients of on either side of our original equation, which gives Since , we can substitute this value to get Multiplying both sides by 2 and then dividing by 7, we deduce that .
The ability to add or subtract vectors through repeated applications of the triangle law for vector addition is extremely useful. In particular, it allows us to write individual vectors in terms of others that are labeled on the same vector diagram. If asked to find a vector , as long as we can find a directed path from point to point (which might require reversing the direction of some vectors along the path by taking their negatives), then will be the resultant sum of all vectors along this directed path.
For example, suppose we are asked to find vector in the diagram below.
As it stands, there is no directed path from to because some of the arrows go the wrong way. However, if we reverse the directions of vectors and by replacing them with their negatives, we create the directed path from to , shown in the next diagram.
Therefore, we get the resultant vector Note that hidden in this calculation are two applications of triangle law for vector addition:
- In , we have .
- In ,
we have .
As and , this becomes as claimed.
The above discussion about the properties of directed paths will help us to solve the next problem.
Example 4: Finding the Sum of Three Vectors in a Triangle
In the triangle , , , and . Which of the following vectors is equal to ?
Answer
We start by drawing a sketch of the triangle described in the question.
We need to find the vector that is equal to .
Since , , and , then Referring to the above sketch and tracing this directed path, we can see that we start at vertex and travel to , then to and finally back to , arriving where we started. Since we start and finish at , the resultant vector is simply , so However, it is obvious that any vector from a point to itself is simply the zero vector, , because the overall effect is that we stay in the same place. Since and , the two right-hand expressions must be equal, giving us
Reviewing the available options, we see that the correct answer is .
By using our method of determining the vector from one point to another by finding a directed path, we can also answer questions involving midpoints of sides or points that are partway along a side. Below is an example of this type.
Example 5: Writing Vectors given as Midpoints and Ratios of Line Segments in a Triangle in Terms of Other Vectors
In triangle below, and .
is the midpoint of and divides in the ratio .
Write the following vectors in terms of and :
Answer
I. To work out , first recall the triangle law for vector addition, which states that when two vectors are represented in magnitude and direction as two sides of a triangle taken in order, the third side of the triangle, directed oppositely, represents the magnitude and direction of their sum.
Observe that in , the vector would be the resultant of and .
Considering , we are given the vector , which goes from to . To go from to , we reverse the direction of this vector to get its negative, so . Furthermore, we are told that .
Therefore, we can apply the triangle law to get
II. Next, we work out . We are told that is the midpoint of the line segment , so , and, therefore, .
Since , this implies that .
III. To work out , we are told in the question that divides in the ratio . This means that point will be four-fifths of the way along line segment , so and, hence, .
As , we conclude that .
IV. Finally, to work out , recall that, in a vector diagram, if we can find a directed path from to , then the vector will be the resultant sum of all vectors along this directed path. We can easily find a suitable directed path from to as shown in the diagram below.
This means we have In part II, we worked out that . Thus, to find , we reverse the direction of this vector to get its negative, so . In addition, from part III we have .
Substituting for and in the above equation, we get
Let us finish by recapping some key points from this explainer.
Key Points
- Vectors are mathematical objects with both magnitude (size) and direction; this is in contrast to scalars, which have only magnitude. Geometrically, we can represent a vector as a directed line segment, where the length of the line segment denotes the magnitude and the orientation of the arrow shows the direction.
- The triangle law for vector addition states that when two vectors are represented in magnitude and direction as two sides of a triangle taken in order, the third side of the triangle, directed oppositely, represents the magnitude and direction of their sum.
- If we reverse the direction of a vector , then we get its negative, . Subtracting a vector is the same as adding its negative.
- When given a vector diagram, as long as we can find a directed path from point to point
(which might require reversing the direction of some vectors along the path by taking their negatives),
we can conclude that the vector will be the resultant sum of all vectors along this directed path.
- The above method also enables us to write vectors given as the midpoints or ratios of line segments in terms of other vectors.
