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Lesson Explainer: Vectors: Geometric Approach Mathematics

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 𝑆𝑇.

For other vectors, we use bold lowercase letters, like a above; this is usually handwritten as 𝑎.

We also meet vectors that combine both sorts of labeling, such as 𝐴𝐵=c 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 𝑃𝑄 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 𝑃𝑄=𝑎, the vector with the same magnitude but opposite direction, which, therefore, has its arrow pointing in the opposite direction, is 𝑄𝑃=𝑃𝑄=𝑎.

In general, for any nonzero scalar 𝑘, the vector 𝑘𝑎 is a vector with |𝑘| times the magnitude of 𝑎. If 𝑘>0, the vector 𝑘𝑎 has the same direction as 𝑎; if 𝑘<0, the vector 𝑘𝑎 has the opposite direction to 𝑎. A few examples are shown below for the cases 𝑘=1, 2, and 12.

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 𝑎 to the tip of vector 𝑏, forming a triangle. It represents the sum 𝑎+𝑏.

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

𝐴𝐵𝐶𝐷 is a parallelogram with 𝐴𝐵=𝑝 and 𝐴𝐷=𝑞.

Find 𝐴𝐶.

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 𝐴𝐵𝐶𝐷, vector 𝑝 is represented as the directed line segment 𝐴𝐵. As side 𝐴𝐵 of the parallelogram is opposite to side 𝐷𝐶, these sides must be parallel and equal in length. Therefore, the corresponding vectors must be parallel and equal in magnitude.

Similarly, vector 𝑞 is represented as the directed line segment 𝐴𝐷. As side 𝐴𝐷 of the parallelogram is opposite to side 𝐵𝐶, 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 𝐴𝐵=𝐷𝐶=𝑝 and 𝐴𝐷=𝐵𝐶=𝑞. 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 𝐴𝐶, note that it is the resultant vector of 𝐴𝐵 and 𝐵𝐶 in 𝐴𝐵𝐶. Therefore, by applying the triangle law, we have 𝐴𝐶=𝐴𝐵+𝐵𝐶=𝑝+𝑞. Equivalently, in 𝐴𝐷𝐶, the vector 𝐴𝐶 is the resultant vector of 𝐴𝐷 and 𝐷𝐶. This time, applying the triangle law, we have 𝐴𝐶=𝐴𝐷+𝐷𝐶=𝑞+𝑝=𝑝+𝑞. 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 𝑎 and 𝑏, as shown below, and we wish to work out 𝑎𝑏.

First, we reverse the direction of 𝑏 to get 𝑏. Then, positioning the head of vector 𝑎 at the tail of vector 𝑏, we can now treat this as an addition. Applying the triangle law for vector addition, we get the resultant vector 𝑎+𝑏=𝑎𝑏.

Note that a special case of this rule occurs when subtracting 𝑎 from 𝑎. Positioning the head of the first vector 𝑎 at the tail of the second vector 𝑎 means that we reverse our direction and travel back to where we started once we reach the end of vector 𝑎. Hence, in this case, the resultant vector is 𝑎+𝑎=𝑎𝑎=0, the zero vector. Another way to think of this operation is that if 𝑎=𝑃𝑄, then 𝑎=𝑄𝑃; so, we go from 𝑃 to 𝑄 and then back from 𝑄 to 𝑃, 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, 𝑉𝑊=𝑎, 𝑉𝑋=𝑏, 𝑌𝑉=𝑐, and 𝑌𝑍=𝑑.

Write the following vectors in terms of 𝑎, 𝑏, 𝑐, and 𝑑:

  1. 𝑍𝑌
  2. 𝑊𝑋
  3. 𝑍𝑉
  4. 𝑍𝑊

Answer

Recall that a vector is a mathematical object that has both magnitude and direction.

I. We start by finding 𝑍𝑌. Note that the diagram features the vector 𝑌𝑍=𝑑, which goes from 𝑌 to 𝑍. Therefore, to go from 𝑍 to 𝑌, we simply reverse the direction of this vector to get its negative, so 𝑍𝑌=𝑑.

II. Next, we must find 𝑊𝑋. 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 𝑊𝑉𝑋, vector 𝑊𝑋 would be the resultant of 𝑊𝑉 and 𝑉𝑋. We are given the vector 𝑉𝑊=𝑎, which goes from 𝑉 to 𝑊. To go from 𝑊 to 𝑉, we reverse the direction of this vector to get its negative, so 𝑊𝑉=𝑎. Moreover, we are also told that 𝑉𝑋=𝑏. Therefore, we can apply the triangle law to get 𝑊𝑋=𝑊𝑉+𝑉𝑋=𝑎+𝑏=𝑏𝑎.

III. To find 𝑍𝑉 notice that in 𝑍𝑌𝑉, vector 𝑍𝑉 would be the resultant of 𝑍𝑌 and 𝑌𝑉. We have 𝑍𝑌=𝑑 from part I, and the question tells us that 𝑌𝑉=𝑐. Applying the triangle law, we have 𝑍𝑉=𝑍𝑌+𝑌𝑉=𝑑+𝑐=𝑐𝑑.

IV. Lastly, we must find 𝑍𝑊. This time, notice that if we were to form the triangle 𝑍𝑉𝑊, as shown below, then vector 𝑍𝑊 would be the resultant of 𝑍𝑉 and 𝑉𝑊.

We already know that 𝑉𝑊=𝑎, and in part III we found that 𝑍𝑉=𝑐𝑑. Therefore, applying the triangle law, we have 𝑍𝑊=𝑍𝑉+𝑉𝑊=𝑐𝑑+𝑎=𝑎+𝑐𝑑.

In our next example, we investigate parallel vectors.

Example 3: Finding an Unknown in a Vector given a Parallel Vector

Suppose 𝑚 and 𝑛 are a pair of nonparallel vectors and that the vectors 4𝑚+𝜆𝑛 and 14𝑚+3𝑛 are parallel. Work out the value of 𝜆.

Answer

Recall that parallel vectors can always be written as nonzero scalar multiples of each other. Hence, in this case, we can write 14𝑚+3𝑛=𝑘4𝑚+𝜆𝑛 for some nonzero scalar 𝑘. Expanding the brackets on the right-hand side, we get the equation 14𝑚+3𝑛=4𝑘𝑚+𝜆𝑘𝑛. We start by working out the value of 𝑘. To do this, we equate the coefficients of 𝑚 on either side of the above equation, which gives 14=4𝑘.

Dividing both sides by 4 and simplifying the resultant fraction, we get 𝑘=144=72.

To find the value of 𝜆, we equate the coefficients of 𝑛 on either side of our original equation, which gives 3=𝜆𝑘. Since 𝑘=72, we can substitute this value to get 3=72𝜆. Multiplying both sides by 2 and then dividing by 7, we deduce that 𝜆=67.

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 𝐴𝐵, as long as we can find a directed path from point 𝐴 to point 𝐵 (which might require reversing the direction of some vectors along the path by taking their negatives), then 𝐴𝐵 will be the resultant sum of all vectors along this directed path.

For example, suppose we are asked to find vector 𝑃𝑆 in the diagram below.

As it stands, there is no directed path from 𝑃 to 𝑆 because some of the arrows go the wrong way. However, if we reverse the directions of vectors 𝑏 and 𝑐 by replacing them with their negatives, we create the directed path from 𝑃 to 𝑆, shown in the next diagram.

Therefore, we get the resultant vector 𝑃𝑆=𝑃𝑄+𝑄𝑅+𝑅𝑆=𝑎+𝑏+𝑐=𝑎𝑏𝑐. Note that hidden in this calculation are two applications of triangle law for vector addition:

  1. In 𝑃𝑄𝑅, we have 𝑃𝑅=𝑃𝑄+𝑄𝑅=𝑎+𝑏=𝑎𝑏.
  2. In 𝑃𝑅𝑆, we have 𝑃𝑆=𝑃𝑅+𝑅𝑆.
    As 𝑃𝑅=𝑎𝑏 and 𝑅𝑆=𝑐, this becomes 𝑃𝑆=𝑎𝑏+𝑐=𝑎𝑏𝑐 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 𝐴𝐵𝐶, 𝐴𝐵=𝑝, 𝐵𝐶=𝑞, and 𝐶𝐴=𝑟. Which of the following vectors is equal to 𝑝+𝑞+𝑟?

  1. 𝐴𝐶
  2. 𝐵𝐴
  3. 0
  4. 𝐶𝐴
  5. 2𝐴𝐶

Answer

We start by drawing a sketch of the triangle described in the question.

We need to find the vector that is equal to 𝑝+𝑞+𝑟.

Since 𝐴𝐵=𝑝, 𝐵𝐶=𝑞, and 𝐶𝐴=𝑟, then 𝑝+𝑞+𝑟=𝐴𝐵+𝐵𝐶+𝐶𝐴. Referring to the above sketch and tracing this directed path, we can see that we start at vertex 𝐴 and travel to 𝐵, then to 𝐶 and finally back to 𝐴, arriving where we started. Since we start and finish at 𝐴, the resultant vector is simply 𝐴𝐴, so 𝑝+𝑞+𝑟=𝐴𝐴. However, it is obvious that any vector from a point to itself is simply the zero vector, 0, because the overall effect is that we stay in the same place. Since 𝐴𝐴=𝑝+𝑞+𝑟 and 𝐴𝐴=0, the two right-hand expressions must be equal, giving us 𝑝+𝑞+𝑟=0.

Reviewing the available options, we see that the correct answer is 0.

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 𝑋𝑌𝑍 below, 𝑋𝑌=𝑎 and 𝑋𝑍=𝑏.

𝑃 is the midpoint of 𝑋𝑌 and 𝑄 divides 𝑋𝑍 in the ratio 41.

Write the following vectors in terms of 𝑎 and 𝑏:

  1. 𝑌𝑍
  2. 𝑋𝑃
  3. 𝑋𝑄
  4. 𝑃𝑄

Answer

I. To work out 𝑌𝑍, 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 𝑌𝑋𝑍, the vector 𝑌𝑍 would be the resultant of 𝑌𝑋 and 𝑋𝑍.

Considering 𝑌𝑋, we are given the vector 𝑋𝑌=𝑎, which goes from 𝑋 to 𝑌. To go from 𝑌 to 𝑋, we reverse the direction of this vector to get its negative, so 𝑌𝑋=𝑎. Furthermore, we are told that 𝑋𝑍=𝑏.

Therefore, we can apply the triangle law to get 𝑌𝑍=𝑌𝑋+𝑋𝑍=𝑎+𝑏=𝑏𝑎.

II. Next, we work out 𝑋𝑃. We are told that 𝑃 is the midpoint of the line segment 𝑋𝑌, so 𝑋𝑃=12𝑋𝑌, and, therefore, 𝑋𝑃=12𝑋𝑌.

Since 𝑋𝑌=𝑎, this implies that 𝑋𝑃=12𝑋𝑌=12𝑎.

III. To work out 𝑋𝑄, we are told in the question that 𝑄 divides 𝑋𝑍 in the ratio 41. This means that point 𝑄 will be four-fifths of the way along line segment 𝑋𝑍, so 𝑋𝑄=45𝑋𝑍 and, hence, 𝑋𝑄=45𝑋𝑍.

As 𝑋𝑍=𝑏, we conclude that 𝑋𝑄=45𝑋𝑍=45𝑏.

IV. Finally, to work out 𝑃𝑄, recall that, in a vector diagram, if we can find a directed path from 𝑃 to 𝑄, then the vector 𝑃𝑄 will be the resultant sum of all vectors along this directed path. We can easily find a suitable directed path from 𝑃 to 𝑄 as shown in the diagram below.

This means we have 𝑃𝑄=𝑃𝑋+𝑋𝑄. In part II, we worked out that 𝑋𝑃=12𝑎. Thus, to find 𝑃𝑋, we reverse the direction of this vector to get its negative, so 𝑃𝑋=12𝑎. In addition, from part III we have 𝑋𝑄=45𝑏.

Substituting for 𝑃𝑋 and 𝑋𝑄 in the above equation, we get 𝑃𝑄=12𝑎+45𝑏=45𝑏12𝑎.

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 𝑎, then we get its negative, 𝑎. 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 𝐴 to point 𝐵 (which might require reversing the direction of some vectors along the path by taking their negatives), we can conclude that the vector 𝐴𝐵 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.

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