We can do vector subtraction just like how we do the subtraction of scalars. We subtract the corresponding components of vectors while subtracting vectors. The graphical interpretation of vector subtraction can be understood by using the parallelogram law and triangle law of addition.
Let us learn more about vector subtraction along with geometrical interpretation and examples.
|1.||What is Vector Subtraction?|
|2.||Vector Subtraction by Parallelogram Law|
|3.||Vector Subtraction by Triangle Law|
|4.||How to Subtract Vectors?|
|5.||Properties of Vector Subtraction|
|6.||FAQs on Vector Subtraction|
What is Vector Subtraction?
The vector subtraction of two vectors a and b is represented by a - b and it is nothing but adding the negative of vector b to the vector a. i.e., a - b = a + (-b). Thus, subtraction of vectors involves the addition of vectors and the negative of a vector. The result of vector subtraction is again a vector. The following are the rules for subtracting vectors:
- It should be performed between two vectors only (not between one vector and one scalar).
- Both vectors in the subtraction should represent the same physical quantity.
Let us understand how to subtract vectors graphically in the upcoming sections.
Vector Subtraction by Parallelogram Law
Suppose that a and b are two vectors. How can we interpret the subtraction of these vectors graphically? That is, what meaning do we attach to a - b? To start with, we note that a - b will be a vector which when added to b should give back a. i.e.,
(a - b) + b = a
But how do we determine the vector a - b, given the vectors a and b? The following figure shows vectors a and b (we have drawn them to be co-initial).
Using the parallelogram law of vector addition, we can determine the vector as follows. We interpret a - b as a + (- b), that is, the vector sum of a and −b. Now, we reverse vector b, and then add a and -b using the parallelogram law:
This shows the vector subtraction a - b as the addition of a and −b.
Vector Subtraction by Triangle Law
Now, we will interpret the subtraction of vectors using the triangle law of vector addition. Denote the vector drawn from the end-point of b to the end-point of a by c.
Note that b + c = a. Thus, c = a - b. In other words, the vector a - b is the vector drawn from the tip of b to the tip of a (if a and b are co-initial).
Note that both ways (using parallelogram law and triangle law) are described above give us the same vector for a - b. This becomes clearer from the figure below:
The vector PT is obtained by adding a and −b using the parallelogram law. The vector RQ is obtained by drawing the vector from the tip of b to the tip of -a. Clearly, both vectors are the same (as their magnitudes and directions are the same).
How to Subtract Vectors?
Here are multiple ways of subtracting vectors:
- To subtract two vectors a and b graphically (i.e., to find a - b), just make them coinitial first and then draw a vector from the tip of b to the tip of a.
- We can add -b (the negative of vector b which is obtained by multiplying b with -1) to a to perform the vector subtraction a - b. i.e., a - b = a + (-b).
- If the vectors are in the component form we can just subtract their respective components in the order of subtraction of vectors.
Here is an example.
Example: If a = <4, -2, 3> and b = <1, -2, 5> then find a - b.
a - b = <4, -2, 3> - <1, -2, 5>
= <4 - 1, -2 - (-2), 3 - 5>
= <3, 0, -2>
Therefore, a - b = <3, 0, -2>.
Properties of Vector Subtraction
Here are some important properties of vector subtraction.
- Any vector subtracted from itself results in a zero vector. i.e., a - a = 0, for any vector a.
- The subtraction of vectors is NOT commutative. i.e., a - b is not necessarily equal to b - a.
- The vector subtraction is NOT associative. i.e., (a - b) - c does not need to be equal to a - (b - c).
- (a - b) · (a + b) = |a|2 - |b|2.
- (a - b) · (a - b) = |a - b|2 = |a|2 + |b|2 - 2 a · b.
☛ Related Topics:
Vector Subtraction Examples
Example 1: Compute the vector subtraction a - b if a = <1, -2, 5> and b = <3, -1, 2>. Also, find its magnitude.
Given that a = <1, -2, 5> and b = <3, -1, 2>.
Now we will find their difference by subtracting the respective components.
a - b = <1, -2, 5> - <3, -1, 2>
= <1 - 3, -2 - (-1), 5 - 2>
= <-2, -1, 3>
Its magnitude is,
|a - b| = √[(-2)2 + (-1)2 + 32]
= √(4 + 1 + 9)
Answer: a - b = <-2, -1, 3> and its magnitude is √14.
Example 2: If θ is the angle between two unit vectors a and b, then find the magnitude of their difference in terms of θ.
From the properties of vector subtraction,
|a - b|2 = |a|2 + |b|2 - 2 a · b ... (1)
By the definition of dot product,
a · b = |a| |b| cos θ
Also, since a and b are unit vectors, |a| = |b| = 1.
Substituting these in (1):
|a - b|2 = |a|2 + |b|2 - 2 |a| |b| cos θ
= 1 + 1 - 2 (1)(1) cos θ
= 2 - 2 cos θ
= 2 (1 - cos θ)
Using trig formulas, 1 - cos θ = 2 sin2 (θ/2).
So |a - b|2 = 2 (2 sin2(θ/2)) = 4 sin2(θ/2).
Taking square root,
|a - b| = 2 sin (θ/2).
Answer: |a - b| = 2 sin (θ/2).
Example 3: What can you say about two vectors a and b if the magnitudes of their sum and difference are equal to each other?
It is given that
|a - b| = |a + b|
Squaring on both sides,
|a - b|2 = |a + b|2
By the properties of vector subtraction,
|a|2 + |b|2 - 2 |a| |b| cos θ = |a|2 + |b|2 + 2 |a| |b| cos θ
4|a| |b| cos θ = 0
cos θ = 0
θ = π/2
Answer: a and b are perpendicular.
FAQs on Vector Subtraction
What is Meant by Vector Subtraction?
Vector subtraction of two vectors a and b is just the addition of vectors a and -b. i.e., a - b = a + (-b). To find -b, we have to multiply each component of vector b by -1.
How Do You Subtract Vectors?
- To subtract a vector b from another vector a, just make them coinitial and then draw a vector from the tip of b to the tip of a.
- To subtract a vector b from another vector a when their components are given, then just subtract every component of b from the corresponding component of a.
How to Find Vector Difference Graphically?
To find the vector difference of two vectors a and b graphically:
- Draw both vectors to start from the same initial point.
- Draw the difference vector a - b from the tip of b to the tip of a.
How to Subtract Vectors in Component Form?
If two vectors are a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>, then their difference is represented by a - b and is obtained by subtracting components of b from the respective components of a. i.e., a - b = <a₁ - b₁, a₂ - b₂, a₃ - b₃>.
What is the Parallelogram Law of Vector Subtraction?
The parallelogram law of subtraction says "if two vectors a and -b are starting from a point P and they are two adjacent sides of a parallelogram, then their sum which is a + (-b) (which can also be written as a - b) is the vector that represents the diagonal of a parallelogram that starts from P".
What is the Triangle Law of Vector Subtraction?
The triangle law of vector subtraction says "to find the difference a - b of two vectors a and b (that are coinitial), just draw a vector from the tail of b to the tail of a".