Let \vec{a}=\left(\begin{array}{l}1 \\ 2 \\ 5\end{array}\right ] And \vec{c}=\left(\begin{array}{c}-1 \\ P \\ Q\end{array}\right ].Find The Values Of P P P And Q Q Q Such That C ⃗ \vec{c} C Is Parallel To

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Introduction

In mathematics, the concept of parallel vectors is crucial in understanding various geometric and algebraic properties. Two vectors are said to be parallel if they are scalar multiples of each other. In this article, we will explore the problem of finding the values of $p$ and $q$ such that $\vec{c}$ is parallel to $\vec{a}$, where \vec{a}=\left(\begin{array}{l}1 \\ 2 \\ 5\end{array}\right] and \vec{c}=\left(\begin{array}{c}-1 \\ p \\ q\end{array}\right].

Understanding Parallel Vectors

To begin with, let's understand what it means for two vectors to be parallel. Two vectors $\vec{u}$ and $\vec{v}$ are said to be parallel if there exists a scalar $k$ such that $\vec{u}=k\vec{v}$. In other words, $\vec{u}$ and $\vec{v}$ are parallel if they have the same direction, but may differ in magnitude.

Conditions for Parallel Vectors

For two vectors to be parallel, the following conditions must be satisfied:

• The vectors must have the same direction.
• The vectors must be scalar multiples of each other.

Finding the Values of $p$ and $q$

To find the values of $p$ and $q$ such that $\vec{c}$ is parallel to $\vec{a}$, we need to satisfy the conditions for parallel vectors. Since \vec{a}=\left(\begin{array}{l}1 \\ 2 \\ 5\end{array}\right] and \vec{c}=\left(\begin{array}{c}-1 \\ p \\ q\end{array}\right], we can write the following equations:

$$\left(\begin{array}{c}-1 \\ p \\ q\end{array}\right)=k\left(\begin{array}{l}1 \\ 2 \\ 5\end{array}\right)$$

Equating Corresponding Components

Equating corresponding components, we get the following equations:

$$-1=k(1)$$

$$p=k(2)$$

$$q=k(5)$$

Solving for $k$

Solving for $k$, we get:

$$k=-1$$

Substituting the Value of $k$

Substituting the value of $k$ into the equations for $p$ and $q$, we get:

$$p=(-1)(2)=-2$$

$$q=(-1)(5)=-5$$

Conclusion

In conclusion, the values of $p$ and $q$ such that $\vec{c}$ is parallel to $\vec{a}$ are $p=-2$ and $q=-5$. This means that the vector $\vec{c}=\left(\begin{array}{c}-1 \\ -2 \\ -5\end{array}\right)$ is parallel to the vector \vec{a}=\left(\begin{array}{l}1 \\ 2 \\ 5\end{array}\right].

Example Use Case

The concept of parallel vectors has numerous applications in mathematics and physics. For instance, in physics, the concept of parallel vectors is used to describe the motion of objects in a straight line. In mathematics, the concept of parallel vectors is used to prove various geometric and algebraic theorems.

Final Thoughts

In this article, we explored the problem of finding the values of $p$ and $q$ such that $\vec{c}$ is parallel to $\vec{a}$. We used the concept of parallel vectors and the conditions for parallel vectors to derive the values of $p$ and $q$. The concept of parallel vectors is a fundamental concept in mathematics and has numerous applications in physics and other fields.

References

• [1] "Vector Calculus" by Michael Corral
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus: Early Transcendentals" by James Stewart

Further Reading

For further reading on the topic of parallel vectors, we recommend the following resources:

• "Vector Calculus" by Michael Corral
• "Linear Algebra and Its Applications" by Gilbert Strang
• "Calculus: Early Transcendentals" by James Stewart

Glossary

• Parallel Vectors: Two vectors that are scalar multiples of each other.
• Scalar Multiple: A vector that is obtained by multiplying a vector by a scalar.
• Scalar: A number that is used to multiply a vector.
• Vector: A mathematical object that has both magnitude and direction.
  

Introduction

In our previous article, we explored the concept of parallel vectors and derived the values of $p$ and $q$ such that $\vec{c}$ is parallel to $\vec{a}$. In this article, we will answer some frequently asked questions (FAQs) on parallel vectors.

Q: What is the difference between parallel and perpendicular vectors?

A: Parallel vectors are scalar multiples of each other, while perpendicular vectors are orthogonal to each other. In other words, parallel vectors have the same direction, while perpendicular vectors have different directions.

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

A: To determine if two vectors are parallel, you can check if they are scalar multiples of each other. If the ratio of the corresponding components of the two vectors is the same, then the vectors are parallel.

Q: What is the significance of parallel vectors in mathematics and physics?

A: Parallel vectors have numerous applications in mathematics and physics. In mathematics, parallel vectors are used to prove various geometric and algebraic theorems. In physics, parallel vectors are used to describe the motion of objects in a straight line.

Q: Can two vectors be parallel and perpendicular at the same time?

A: No, two vectors cannot be parallel and perpendicular at the same time. If two vectors are parallel, they have the same direction, while if they are perpendicular, they have different directions.

Q: How do I find the values of $p$ and $q$ such that $\vec{c}$ is parallel to $\vec{a}$?

A: To find the values of $p$ and $q$ such that $\vec{c}$ is parallel to $\vec{a}$, you can use the concept of parallel vectors and the conditions for parallel vectors. You can equate corresponding components and solve for $p$ and $q$.

Q: What is the relationship between parallel vectors and linear dependence?

A: Parallel vectors are linearly dependent, while non-parallel vectors are linearly independent. In other words, if two vectors are parallel, they can be expressed as scalar multiples of each other, while if they are non-parallel, they cannot be expressed as scalar multiples of each other.

Q: Can a vector be parallel to itself?

A: Yes, a vector can be parallel to itself. In fact, a vector is always parallel to itself, since it is a scalar multiple of itself.

Q: How do I visualize parallel vectors in 3D space?

A: To visualize parallel vectors in 3D space, you can use a 3D coordinate system and plot the vectors on the coordinate plane. You can also use a vector diagram to visualize the parallel vectors.

Q: What is the importance of parallel vectors in computer graphics?

A: Parallel vectors are used extensively in computer graphics to describe the motion of objects in 3D space. They are used to create realistic animations and simulations.

Q: Can parallel vectors be used to solve real-world problems?

A: Yes, parallel vectors can be used to solve real-world problems. They are used in various fields such as physics, engineering, and computer science to describe the motion of objects and solve problems.

Q: How do I apply the concept of parallel vectors to real-world problems?

A: To apply the concept of parallel vectors to real-world problems, you can use the conditions for parallel vectors and the concept of scalar multiples. You can also use vector diagrams and 3D coordinate systems to visualize the parallel vectors.

Conclusion

In conclusion, parallel vectors are a fundamental concept in mathematics and physics. They have numerous applications in various fields and can be used to solve real-world problems. We hope that this FAQ article has provided you with a better understanding of parallel vectors and their significance in mathematics and physics.

References

• [1] "Vector Calculus" by Michael Corral
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus: Early Transcendentals" by James Stewart

Further Reading

For further reading on the topic of parallel vectors, we recommend the following resources:

• "Vector Calculus" by Michael Corral
• "Linear Algebra and Its Applications" by Gilbert Strang
• "Calculus: Early Transcendentals" by James Stewart

Glossary

• Parallel Vectors: Two vectors that are scalar multiples of each other.
• Scalar Multiple: A vector that is obtained by multiplying a vector by a scalar.
• Scalar: A number that is used to multiply a vector.
• Vector: A mathematical object that has both magnitude and direction.