Definition of Coplanar Vectors

Oct 13, 2025 by miodeya.com 52 views

8. Prove that the following vectors are coplanar

In this section, we will prove that the given vectors are coplanar. To do this, we will use the concept of linear dependence and show that the vectors can be expressed as linear combinations of each other.

Definition of Coplanar Vectors

Two or more vectors are said to be coplanar if they lie in the same plane. In other words, if the vectors can be expressed as linear combinations of each other, then they are coplanar.

Part a: Prove that the vectors ${ $2 \vec{a} - 3 \vec{b} + 4 \vec{c}\$}$ , ${$ -\vec{a} + 3 \vec{b} - 5 \vec{c}$}$, and ${$ -\vec{a} + 2 \vec{b} - 3 \vec{c}$}$ are coplanar

To prove that the vectors are coplanar, we need to show that they can be expressed as linear combinations of each other. Let's start by assuming that the vectors are not coplanar. Then, we can write the following equation:

\begin{vmatrix} 2 & -1 & -1 \\ -3 & 3 & 2 \\ 4 & -5 & -3 \end{vmatrix} \neq 0

where the determinant is calculated using the vectors ${ $2 \vec{a} - 3 \vec{b} + 4 \vec{c}\$}$ , ${$ -\vec{a} + 3 \vec{b} - 5 \vec{c}$}$, and ${$ -\vec{a} + 2 \vec{b} - 3 \vec{c}$}$.

However, if we expand the determinant, we get:

\begin{vmatrix} 2 & -1 & -1 \\ -3 & 3 & 2 \\ 4 & -5 & -3 \end{vmatrix} = 2(9+10) - (-1)(-9-8) -1(-15+12)

= 2(19) - (-1)(-17) -1(-3)

= 38 - 17 + 3

= 24

Since the determinant is equal to 24, which is not equal to 0, we can conclude that the vectors are coplanar.

Part b: Prove that the vectors ${$ \vec{a} + \vec{b} + \vec{c}$}$ are coplanar

\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} \neq 0

where the determinant is calculated using the vectors ${$ \vec{a} + \vec{b} + \vec{c}$}$.

However, if we expand the determinant, we get:

\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} = 1(1-1) - 1(1-1) + 1(1-1)

= 1(0) - 1(0) + 1(0)

= 0

Since the determinant is equal to 0, we can conclude that the vectors are not coplanar.

In this section, we proved that the given vectors are coplanar. We used the concept of linear dependence and showed that the vectors can be expressed as linear combinations of each other. We also used the determinant to prove that the vectors are coplanar.

Key Takeaways

Two or more vectors are said to be coplanar if they lie in the same plane.
If the vectors can be expressed as linear combinations of each other, then they are coplanar.
We can use the determinant to prove that the vectors are coplanar.

References

Linear Algebra and Its Applications
Determinants and Matrices
Coplanar Vectors and Linear Dependence
Q&A: Coplanar Vectors =========================

Frequently Asked Questions

In this section, we will answer some of the most frequently asked questions about coplanar vectors.

Q: What are coplanar vectors?

A: Coplanar vectors are two or more vectors that lie in the same plane. In other words, if the vectors can be expressed as linear combinations of each other, then they are coplanar.

Q: How do I determine if two vectors are coplanar?

A: To determine if two vectors are coplanar, you can use the concept of linear dependence. If the vectors can be expressed as linear combinations of each other, then they are coplanar. You can also use the determinant to prove that the vectors are coplanar.

Q: What is the difference between coplanar and linearly independent vectors?

A: Coplanar vectors are vectors that lie in the same plane, while linearly independent vectors are vectors that cannot be expressed as linear combinations of each other. In other words, coplanar vectors are a special case of linearly independent vectors.

Q: Can three vectors be coplanar?

A: Yes, three vectors can be coplanar. In fact, three vectors are coplanar if and only if the determinant of the matrix formed by the vectors is equal to 0.

Q: Can four vectors be coplanar?

A: No, four vectors cannot be coplanar. In fact, four vectors are linearly dependent if and only if the determinant of the matrix formed by the vectors is equal to 0.

Q: What is the significance of coplanar vectors in real-world applications?

A: Coplanar vectors have many real-world applications, including:

Computer graphics: Coplanar vectors are used to create 3D models and animations.
Engineering: Coplanar vectors are used to design and analyze mechanical systems.
Physics: Coplanar vectors are used to describe the motion of objects in space.

Q: How do I prove that a set of vectors is coplanar?

A: To prove that a set of vectors is coplanar, you can use the following steps:

Check for linear dependence: Check if the vectors can be expressed as linear combinations of each other.
Calculate the determinant: Calculate the determinant of the matrix formed by the vectors.
Check if the determinant is equal to 0: If the determinant is equal to 0, then the vectors are coplanar.

In this section, we answered some of the most frequently asked questions about coplanar vectors. We hope that this article has provided you with a better understanding of coplanar vectors and how to determine if a set of vectors is coplanar.

Key Takeaways

Coplanar vectors are two or more vectors that lie in the same plane.
To determine if two vectors are coplanar, you can use the concept of linear dependence.
The determinant can be used to prove that a set of vectors is coplanar.