Collinear Vectors: Finding P And Q Parameters

Hey guys! Today, we're diving into the fascinating world of vectors, specifically focusing on collinear vectors. We'll be tackling the challenge of finding the values of parameters p and q that make given vectors collinear. This is a super important concept in geometry and linear algebra, so let's break it down together. We will solve the following problems:

• 4. 15. $\vec{a} = p\vec{i} +2\vec{j}-4\vec{k}$, $\vec{b} =-2\vec{i}+q\vec{j}+8\vec{k}$.
• 4. 16. $\vec{a}=3\vec{i}+p\vec{j}+6\vec{k}$, $\vec{b} =-\vec{i}-2\vec{j}+q\vec{k}$.

Understanding Collinear Vectors

First off, what does it mean for vectors to be collinear? Collinear vectors are vectors that lie on the same line or on parallel lines. Think of it like this: they're either pointing in the exact same direction, the exact opposite direction, or they're parallel. Mathematically, this means one vector can be obtained by multiplying the other vector by a scalar (a constant number). This scalar can be positive, negative, or even zero. Understanding this fundamental concept is crucial before we jump into solving problems. Essentially, if vectors $\vec{a}$ and $\vec{b}$ are collinear, then there exists a scalar k such that $\vec{a} = k\vec{b}$. This relationship forms the backbone of our problem-solving approach. When we say vectors lie on the same line, we are essentially talking about their directional properties. The magnitude might differ, but the direction remains the same or exactly opposite, making them scalar multiples of each other. This scalar relationship allows us to set up equations and solve for unknown parameters, making it a powerful tool in vector analysis. Furthermore, the concept of collinearity is not just limited to two dimensions; it extends seamlessly into three and higher dimensions. This makes it a versatile concept with applications across various fields of mathematics and physics. In essence, collinearity is a statement about the linear dependence of vectors, highlighting their fundamental relationship within a vector space.

Problem 4.15: $\vec{a} = p\vec{i} +2\vec{j}-4\vec{k}$, $\vec{b} =-2\vec{i}+q\vec{j}+8\vec{k}$

Let's get our hands dirty with the first problem. We have two vectors, $\vec{a}$ and $\vec{b}$, and we need to find the values of p and q that make them collinear. Remember our key concept: if $\vec{a}$ and $\vec{b}$ are collinear, then $\vec{a} = k\vec{b}$ for some scalar k. Let's write this out explicitly:

$p\vec{i} + 2\vec{j} - 4\vec{k} = k(-2\vec{i} + q\vec{j} + 8\vec{k})$

Now, we can distribute the scalar k on the right side:

$p\vec{i} + 2\vec{j} - 4\vec{k} = -2k\vec{i} + kq\vec{j} + 8k\vec{k}$

For these two vectors to be equal, their corresponding components must be equal. This gives us a system of three equations:

1. p = -2k
2. 2 = kq
3. -4 = 8k

Let's solve this system. From equation (3), we can easily find k:

k = -4 / 8 = -1/2

Now that we know k, we can substitute it into equation (1) to find p:

p = -2 * (-1/2) = 1

And finally, we can substitute k into equation (2) to find q:

2 = (-1/2) * q

q = 2 / (-1/2) = -4

So, the values of p and q that make the vectors collinear are p = 1 and q = -4. This demonstrates the power of using the scalar multiple relationship to solve for unknown parameters in vector problems. By breaking down the vector equation into component equations, we can leverage algebraic techniques to arrive at the solution. This method is not only effective but also provides a clear and systematic approach to tackling similar problems. Moreover, understanding the underlying concept of collinearity allows us to visualize the vectors and their relationship, making the solution process more intuitive. In this case, we've successfully determined the values of p and q that ensure the vectors lie on the same line or are parallel, showcasing the practical application of vector algebra in geometric problems.

Problem 4.16: $\vec{a}=3\vec{i}+p\vec{j}+6\vec{k}$, $\vec{b} =-\vec{i}-2\vec{j}+q\vec{k}$

Alright, let's move on to problem 4.16! We're following the same game plan here. We've got two new vectors, $\vec{a}$ and $\vec{b}$, and our mission is to find the values of p and q that make them collinear. Remember the golden rule: if $\vec{a}$ and $\vec{b}$ are collinear, there's a scalar k such that $\vec{a} = k\vec{b}$. Let's write it out:

$3\vec{i} + p\vec{j} + 6\vec{k} = k(-\vec{i} - 2\vec{j} + q\vec{k})$

Distribute that k, guys:

$3\vec{i} + p\vec{j} + 6\vec{k} = -k\vec{i} - 2k\vec{j} + kq\vec{k}$

Now, equate the corresponding components to get our system of equations:

1. 3 = -k
2. p = -2k
3. 6 = kq

Let's solve this bad boy. From equation (1), we get k directly:

k = -3

Substitute k into equation (2) to find p:

p = -2 * (-3) = 6

And finally, substitute k into equation (3) to find q:

6 = (-3) * q

q = 6 / (-3) = -2

So, for these vectors to be collinear, p needs to be 6 and q needs to be -2. This problem further solidifies our understanding of how to apply the scalar multiple relationship in solving for unknown parameters. The systematic approach of setting up component equations and solving for the unknowns proves to be a reliable method. Moreover, the consistency in the solution process across different problems reinforces the underlying principles of vector collinearity. By carefully applying the definition and algebraic techniques, we can confidently tackle a wide range of vector problems. Furthermore, the ability to visualize these vectors and their relationship adds another layer of understanding to the solution. In this case, we've successfully determined the values of p and q that ensure the vectors align on the same line or are parallel, highlighting the practical application of vector algebra in geometric analysis. This skill is invaluable in various fields, from computer graphics to physics, where understanding spatial relationships is crucial.

Key Takeaways

Let's recap what we've learned today, guys:

• Collinear vectors are scalar multiples of each other. This is the most important concept to remember.
• To find parameters that make vectors collinear, set up the equation $\vec{a} = k\vec{b}$, distribute the scalar, and equate corresponding components.
• Solve the resulting system of equations to find the unknown parameters.

By mastering these steps, you'll be able to confidently tackle problems involving collinear vectors. Remember, practice makes perfect! Try working through similar problems to solidify your understanding. The ability to determine collinearity is fundamental in vector algebra and has wide-ranging applications in various scientific and engineering disciplines. By internalizing the concept and the problem-solving approach, you'll be well-equipped to handle more complex vector-related challenges. Furthermore, understanding collinearity lays the groundwork for exploring concepts like linear dependence and independence, which are crucial in linear algebra and higher-level mathematics. So, keep practicing, keep exploring, and you'll become a vector whiz in no time!

I hope this breakdown helps you guys understand how to find the parameters that make vectors collinear. Keep practicing, and you'll be vector pros in no time! Happy vectoring!