Lesson Explainer: Vector Subtraction | Nagwa Lesson Explainer: Vector Subtraction | Nagwa

Lesson Explainer: Vector Subtraction Physics

In this explainer, we will learn how to subtract one vector from another in two dimensions, using both graphical and algebraic methods.

Consider the vectors 𝐴 and 𝐵 shown on the grid below.

Recall that we can work out the result of 𝐴+𝐵 graphically by moving vector 𝐵 so that the “tail” of vector 𝐵 is positioned at the “tip” of vector 𝐴. This is shown on the grid below, and the result of adding the two vectors is 𝑉.

But what would the result of subtracting  𝐵 from 𝐴 be? What would the result of 𝐴𝐵 be?

Recall that the negative of a vector is a vector that has the same magnitude but the opposite direction. The diagram below shows a vector, 𝐵, and its negative, 𝐵.

With ordinary numbers, subtracting a number is the same as adding the negative of that number. So, for example, 124 is the same as 12+(4).

The same is true of vectors: 𝐴𝐵 is the same as 𝐴+𝐵.

So, we can subtract one vector from another by first finding the negative of the vector we want to subtract and then adding the two vectors.

Going back to the first diagram, let’s change 𝐵 to 𝐵.

We can now simply add these two vectors together by placing the tail of 𝐵 on the tip of 𝐴. The result is vector 𝑉 shown on the grid below.

This is one way of subtracting vectors graphically, but there is another way, which is generally faster. Consider again the two vectors 𝐴 and 𝐵, shown on the grid below.

The result of 𝐴𝐵 is a vector that goes from the tip of the second vector (in this case 𝐵) to the tip of the first vector (in this case 𝐴). This is vector 𝑉, shown on the grid below.

If we think of vectors 𝐴 and 𝐵 as being position vectors of points 𝐴 and 𝐵, then the result of 𝐴𝐵 can be thought of as a vector that goes “from  𝐵  to  𝐴”, as shown below.

Notice that, because of this, the result of 𝐴𝐵 is not equal to the result of 𝐵𝐴. The result of 𝐵𝐴 is a vector that goes from 𝐴 to 𝐵, as shown below.

As we can see from the diagram, changing the order of the two vectors in the subtraction produces the negative result. We can write this algebraically as 𝐵𝐴=𝐴𝐵.

Example 1: Subtracting Vectors Graphically

The diagram shows seven vectors: 𝐴, 𝐵, 𝑃, 𝑄, 𝑅, 𝑆, and 𝑇.

  1. Which of the vectors is equal to 𝐴𝐵?
  2. Which of the vectors is equal to 𝐵𝐴?

Answer

Part 1

Recall that we can find the result of 𝐴𝐵 by drawing a vector that goes from the tip of the second vector, 𝐵, to the tip of the first vector, 𝐴. This is shown on the grid below.

Comparing this to the other vectors shown in the diagram in the question, we see that the result of 𝐴𝐵 corresponds to vector 𝑄.

Part 2

Recall that when we reverse the order of the vectors in a vector subtraction, we get the negative result. Expressed algebraically, 𝐵𝐴=𝐴𝐵.

So, we can use our answer from part 1 to quickly work out the answer to this part. If 𝐴𝐵=𝑄, then 𝐵𝐴=(𝑄)𝐵𝐴=𝑄.

Looking back to the diagram in the question, we can see that the vector that is the negative of vector 𝑄 is vector 𝑆.

Example 2: Subtracting Vectors Graphically

Which of the vectors, 𝑃, 𝑄, 𝑅, 𝑆, or 𝑇, shown in the diagram is equal to 𝐴𝐵?

Answer

Recall that we can find the result of 𝐴𝐵 by drawing a vector that goes from the tip of the second vector, 𝐵, to the tip of the first vector, 𝐴. This is shown on the grid below.

Comparing this to the other vectors shown in the diagram in the question, we see that the result of 𝐴𝐵 corresponds to vector 𝑃.

So far, we have looked at how to subtract vectors graphically, but we can also do this algebraically.

Recall that any vector 𝐴 can be represented as a sum of multiples of unit vectors along the 𝑥 and 𝑦 axes. If the horizontal component of 𝐴 is 𝐴, the vertical component of 𝐴 is 𝐴, 𝑖 is a unit vector along the 𝑥-axis, and 𝑗 is a unit vector along the 𝑦-axis, then 𝐴=𝐴𝑖+𝐴𝑗.

Similarly, if we say that 𝐵=𝐵𝑖+𝐵𝑗, then 𝐴𝐵=𝐴𝑖+𝐴𝑗𝐵𝑖+𝐵𝑗.

If we expand out the brackets, we get 𝐴𝐵=𝐴𝑖+𝐴𝑗𝐵𝑖𝐵𝑗.

We can then gather like terms to get 𝐴𝐵=(𝐴𝐵)𝑖+𝐴𝐵𝑗.

So, if we know the components of two vectors, we can use this formula to subtract one vector from the other, and get the components of the resulting vector.

Example 3: Subtracting Two Vectors given Their Components

Consider the two vectors 𝐴 and 𝐵, where 𝐴=8𝑖+10𝑗 and 𝐵=3𝑖+2𝑗. Calculate 𝐴𝐵.

Answer

We can use the formula 𝐴𝐵=(𝐴𝐵)𝑖+𝐴𝐵𝑗 to find the result of 𝐴𝐵, where 𝐴 and 𝐵 are the horizontal components of the two vectors and 𝐴 and 𝐵 are the vertical components of the two vectors.

Substituting in the values given in the question, we get 𝐴𝐵=(83)𝑖+(102)𝑗𝐴𝐵=5𝑖+8𝑗.

The result of 𝐴𝐵 is 5𝑖+8𝑗.

Example 4: Subtracting Two Vectors given Their Components

Consider the two vectors 𝐴 and 𝐵. 𝐴=4𝑖9𝑗 and 𝐵=1𝑖12𝑗. Which of the five vectors shown in the diagram is equal to 𝐴𝐵?

Answer

In this question, we have to work out the result of the vector subtraction algebraically and then identify the resulting vector in the diagram.

We can use the formula 𝐴𝐵=(𝐴𝐵)𝑖+𝐴𝐵𝑗 to find the result of 𝐴𝐵, where 𝐴 and 𝐵 are the horizontal components of the two vectors and 𝐴 and 𝐵 are the vertical components of the two vectors.

Substituting in the values given in the question, we get 𝐴𝐵=(41)𝑖+((9)(12))𝑗𝐴𝐵=(41)𝑖+(9+12)𝑗𝐴𝐵=3𝑖+3𝑗.

The result of 𝐴𝐵 is 3𝑖+3𝑗. Looking at the diagram, we can see that this corresponds to vector 𝑄.

Key Points

  • For two vectors 𝐴 and 𝐵, we can work out the result of 𝐴𝐵 graphically by first finding the negative of 𝐵 and then simply adding it to 𝐴.
  • Another way we can think about doing vector subtraction graphically is to draw a new vector that goes from the “tip” of vector 𝐵 to the “tip” of vector 𝐴.
  • If we are given 𝐴 and 𝐵 in component form, we can use the formula 𝐴𝐵=(𝐴𝐵)𝑖+𝐴𝐵𝑗 to find the result of 𝐴𝐵.

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