Finding the Difference Between Two Vectors

Oct 13, 2025 by wbrum.org 51 views

Introduction

In mathematics, vectors are used to represent quantities with both magnitude and direction. When dealing with vectors, it's often necessary to find the difference between two vectors, which is a fundamental operation in vector algebra. In this article, we will explore how to find the difference between two vectors, using the given vectors $$u =\langle-4,-4\rangle$$ and $$v =\langle-5,6\rangle$$ as examples.

What is the Difference Between Two Vectors?

The difference between two vectors is a new vector that represents the result of subtracting one vector from another. This operation is denoted by the symbol $$-$$, and it's used to find the vector that points from the tip of one vector to the tip of another.

Finding the Difference Between Two Vectors

To find the difference between two vectors, we need to subtract the corresponding components of the two vectors. In other words, we need to subtract the x-component of one vector from the x-component of the other vector, and the y-component of one vector from the y-component of the other vector.

Calculating the Difference Between the Given Vectors

Let's calculate the difference between the given vectors $$u =\langle-4,-4\rangle$$ and $$v =\langle-5,6\rangle$$. To do this, we need to subtract the corresponding components of the two vectors.

v - u = \langle -5 - (-4), 6 - (-4) \rangle

v - u = \langle -5 + 4, 6 + 4 \rangle

v - u = \langle -1, 10 \rangle

Representing the Difference Graphically

Now that we have found the difference between the two vectors, we can represent it graphically in two different ways.

Graphical Representation 1: Vector Diagram

We can represent the difference between the two vectors graphically using a vector diagram. In this diagram, we will draw the two vectors $$u$$ and $$v$$, and then draw the vector $$v - u$$, which represents the difference between the two vectors.

u = (-4, -4)
v = (-5, 6)
v - u = (-1, 10)

Graphical Representation 2: Coordinate Plane

We can also represent the difference between the two vectors graphically using a coordinate plane. In this representation, we will plot the two vectors $$u$$ and $$v$$, and then plot the vector $$v - u$$, which represents the difference between the two vectors.

u = (-4, -4)
v = (-5, 6)
v - u = (-1, 10)

Components of the Difference

The difference between the two vectors $$v - u$$ has two components: the x-component and the y-component. The x-component of the difference is $$-1$$, and the y-component of the difference is $$10$$.

Discussion

In this article, we have explored how to find the difference between two vectors, using the given vectors $$u =\langle-4,-4\rangle$$ and $$v =\langle-5,6\rangle$$ as examples. We have also represented the difference graphically in two different ways, using a vector diagram and a coordinate plane. The difference between two vectors is a fundamental operation in vector algebra, and it has many applications in mathematics and science.

Conclusion

In conclusion, finding the difference between two vectors is a simple operation that involves subtracting the corresponding components of the two vectors. The difference between two vectors can be represented graphically in two different ways, using a vector diagram and a coordinate plane. The components of the difference are the x-component and the y-component, which can be used to represent the difference between the two vectors.

References

[1] "Vector Algebra" by Michael Artin
[2] "Linear Algebra and Its Applications" by Gilbert Strang
[3] "Vector Calculus" by Michael Spivak

Introduction

In our previous article, we explored how to find the difference between two vectors, using the given vectors $$u =\langle-4,-4\rangle$$ and $$v =\langle-5,6\rangle$$ as examples. In this article, we will answer some of the most frequently asked questions about finding the difference between two vectors.

Q&A

Q: What is the difference between two vectors?

A: The difference between two vectors is a new vector that represents the result of subtracting one vector from another. This operation is denoted by the symbol $$-$$, and it's used to find the vector that points from the tip of one vector to the tip of another.

Q: How do I find the difference between two vectors?

A: To find the difference between two vectors, you need to subtract the corresponding components of the two vectors. In other words, you need to subtract the x-component of one vector from the x-component of the other vector, and the y-component of one vector from the y-component of the other vector.

Q: What are the components of the difference between two vectors?

A: The components of the difference between two vectors are the x-component and the y-component. The x-component of the difference is the result of subtracting the x-component of one vector from the x-component of the other vector, and the y-component of the difference is the result of subtracting the y-component of one vector from the y-component of the other vector.

Q: Can I represent the difference between two vectors graphically?

A: Yes, you can represent the difference between two vectors graphically using a vector diagram or a coordinate plane. In a vector diagram, you will draw the two vectors and then draw the vector that represents the difference between the two vectors. In a coordinate plane, you will plot the two vectors and then plot the vector that represents the difference between the two vectors.

Q: What are some real-world applications of finding the difference between two vectors?

A: Finding the difference between two vectors has many real-world applications, including:

Calculating the distance between two points in a coordinate plane
Finding the shortest path between two points in a coordinate plane
Calculating the velocity of an object
Finding the acceleration of an object

Q: Can I use a calculator to find the difference between two vectors?

A: Yes, you can use a calculator to find the difference between two vectors. Most calculators have a built-in function for finding the difference between two vectors, which is denoted by the symbol $$-$$.

Q: What are some common mistakes to avoid when finding the difference between two vectors?

A: Some common mistakes to avoid when finding the difference between two vectors include:

Subtracting the wrong components of the two vectors
Not using the correct operation (subtraction) to find the difference between the two vectors
Not representing the difference between the two vectors graphically

Conclusion

In conclusion, finding the difference between two vectors is a fundamental operation in vector algebra that has many real-world applications. By understanding how to find the difference between two vectors, you can solve a wide range of problems in mathematics and science.

References

[1] "Vector Algebra" by Michael Artin
[2] "Linear Algebra and Its Applications" by Gilbert Strang
[3] "Vector Calculus" by Michael Spivak