Vector P Coordinates And Length Calculation

Hey guys! Today, we're diving into a fun little problem involving vectors. Specifically, we're going to figure out how to find the coordinates and length of a vector when it's expressed as a combination of other vectors. Don't worry, it's not as scary as it sounds! We'll break it down step by step, so you can follow along easily. Let's jump right in!

Understanding the Problem

So, the problem we're tackling today involves finding the coordinates and length of a vector ${\vec{p}}$. This vector isn't just floating in space; it's defined in terms of two other vectors, ${\vec{a}}$ and ${\vec{b}}$. We're given the following information:

• ${\vec{p} = 5\vec{a} - 6\vec{b}}$
• ${\vec{a} = \{5; -5\}}$
• ${\vec{b} = \{5; -4\}}$

In simpler terms, vector ${\vec{p}}$ is a result of scaling vectors ${\vec{a}}$ and ${\vec{b}}$ by 5 and -6 respectively, and then adding them together. We also know the coordinates of vectors ${\vec{a}}$ and ${\vec{b}}$. Our mission, should we choose to accept it (and we do!), is to find the coordinates of ${\vec{p}}$ and its length (also known as its magnitude). Remember, the coordinates of a vector tell us how far to move along the x and y axes (in a 2D plane) to get from the starting point to the endpoint of the vector. The length of a vector, on the other hand, is the straight-line distance between its starting and ending points. To kick things off, let's talk about vector operations. Understanding how to add and scale vectors is the bedrock of solving this problem. When you multiply a vector by a scalar (a regular number), you're essentially stretching or shrinking the vector. A positive scalar will keep the vector pointing in the same direction, while a negative scalar will flip it around. And when you add vectors, you're essentially placing them head-to-tail and finding the resultant vector that connects the starting point of the first vector to the endpoint of the last one. These operations are performed component-wise, which means you're adding or multiplying the corresponding x and y coordinates. This will become clearer as we work through the problem, so don't worry if it sounds a bit abstract right now.

Step 1: Calculate 5a

The first thing we need to do is find out what 5 times vector ${\vec{a}}$ is. This is a scalar multiplication, which means we multiply each component of ${\vec{a}}$ by 5. So, if ${\vec{a} = \{5; -5\}}$, then:

${ 5\vec{a} = 5 \times \{5; -5\} = \{5 \times 5; 5 \times -5\} = \{25; -25\} }$

Simple enough, right? We just multiplied each coordinate by 5. Now we know that 5${\vec{a}}$ is the vector with coordinates (25, -25). Think of it this way: we've taken the original vector ${\vec{a}}$ and stretched it out five times its original length, keeping it pointed in the same direction. This is a fundamental operation in vector algebra, and it's crucial for understanding how vectors combine and interact. You can imagine this graphically as extending the arrow representing ${\vec{a}}$ five times further from the origin. Each component of the vector is scaled proportionally, ensuring that the direction remains consistent. This concept is not only important for solving this specific problem but also for more complex vector manipulations in physics, engineering, and computer graphics. For instance, in computer graphics, scaling vectors is a common operation when resizing objects or adjusting their positions in 3D space. In physics, it might represent increasing the force applied in a particular direction. So, mastering scalar multiplication is a crucial step in building your understanding of vectors and their applications. Now that we've successfully calculated 5${\vec{a}}$, we're one step closer to finding ${\vec{p}}$. The next step involves a similar process, but this time we'll be scaling vector ${\vec{b}}$ by a different scalar. This will further contribute to our understanding of how vector components change under scalar multiplication and how these changes affect the overall vector sum. So, let's move on to the next step and see what happens when we multiply ${\vec{b}}$ by -6.

Step 2: Calculate -6b

Next up, we need to calculate -6 times vector ${\vec{b}}$. Just like before, this is scalar multiplication, but this time we're multiplying by a negative number. Remember, a negative scalar will not only change the magnitude (length) of the vector but also flip its direction. We have ${\vec{b} = \{5; -4\}}$, so:

${ -6\vec{b} = -6 \times \{5; -4\} = \{-6 \times 5; -6 \times -4\} = \{-30; 24\} }$

Okay, we've got -6${\vec{b}}$, which is the vector (-30, 24). Notice how the negative sign in -6 flipped the signs of the components in ${\vec{b}}$. The x-component went from 5 to -30, and the y-component went from -4 to 24. This direction change is a key aspect of scalar multiplication with negative scalars. To visualize this, imagine the original vector ${\vec{b}}$. Multiplying it by -6 stretches it out six times its original length, but also rotates it 180 degrees. This rotation is what the sign flip in the components represents. Understanding this directional change is crucial in many applications. For example, in physics, if ${\vec{b}}$ represents a force, then -6${\vec{b}}$ would represent a force six times stronger acting in the opposite direction. In computer graphics, flipping the direction of a vector can be used to create reflections or reverse animations. Scalar multiplication with negative scalars is a powerful tool, and it's essential to grasp how it affects both the magnitude and direction of a vector. Now that we've calculated both 5${\vec{a}}$ and -6${\vec{b}}$, we're ready to put them together. The next step involves adding these two scaled vectors to find ${\vec{p}}$. This addition will demonstrate how vector components combine to form a resultant vector, bringing us closer to our final goal of finding the coordinates and length of ${\vec{p}}$. So, let's move on and see how vector addition works in practice.

Step 3: Calculate p = 5a - 6b

Now for the main event: finding vector ${\vec{p}}$. We know that ${\vec{p} = 5\vec{a} - 6\vec{b}}$, and we've already calculated 5${\vec{a}}$ and -6${\vec{b}}$. So, all we need to do is add them together. Remember, vector addition is done component-wise, meaning we add the corresponding x-components and the corresponding y-components.

We have:

• 5${\vec{a}}$ = {25; -25}
• -6${\vec{b}}$ = {-30; 24}

Adding these together gives us:

${ \vec{p} = 5\vec{a} - 6\vec{b} = \{25; -25\} + \{-30; 24\} = \{25 + (-30); -25 + 24\} = \{-5; -1\} }$

So, the coordinates of vector ${\vec{p}}$ are (-5, -1). We've found the first part of our solution! Vector addition is a fundamental operation, and it's important to understand how it works both algebraically and geometrically. Algebraically, as we've seen, you simply add the corresponding components. Geometrically, you can visualize vector addition by placing the tail of the second vector at the head of the first vector. The resultant vector (in this case, ${\vec{p}}$) is the vector that goes from the tail of the first vector to the head of the second vector. This head-to-tail method provides a visual representation of how vectors combine to produce a new vector. Understanding this geometric interpretation can be very helpful in solving problems involving forces, velocities, and other physical quantities. For example, if two forces are acting on an object, you can find the net force by adding the force vectors together. The resultant vector will tell you the direction and magnitude of the overall force acting on the object. Now that we've found the coordinates of ${\vec{p}}$, we're just one step away from completing our mission. The final step is to calculate the length (or magnitude) of ${\vec{p}}$. This will involve using the Pythagorean theorem, which provides a direct link between the components of a vector and its length. So, let's move on to the final step and see how to calculate the magnitude of ${\vec{p}}$.

Step 4: Calculate the Length of p

Alright, the final step! We need to find the length (or magnitude) of vector ${\vec{p}}$. The length of a vector is essentially the distance from its starting point (usually the origin) to its endpoint. We can calculate this using the Pythagorean theorem. If ${\vec{p} = \{x; y\}}$, then the length of ${\vec{p}}$, denoted as |${\vec{p}}$|, is given by:

${ |\vec{p}| = \sqrt{x^2 + y^2} }$

We know that ${\vec{p} = \{-5; -1\}}$, so x = -5 and y = -1. Plugging these values into the formula, we get:

${ |\vec{p}| = \sqrt{(-5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} }$

Therefore, the length of vector ${\vec{p}}$ is ${\sqrt{26}}$. And there you have it! We've successfully calculated both the coordinates and the length of vector ${\vec{p}}$. The Pythagorean theorem is a cornerstone of vector calculations, and it's essential to understand its connection to the length of a vector. It's a direct application of the familiar geometric principle to the vector world. You can visualize this by drawing a right triangle where the vector ${\vec{p}}$ is the hypotenuse, and the x and y components are the legs. The Pythagorean theorem then simply states that the square of the hypotenuse (the length of ${\vec{p}}$) is equal to the sum of the squares of the legs (the x and y components). This relationship is fundamental in many areas of mathematics and physics. For example, it's used to calculate distances in coordinate systems, find the magnitude of forces and velocities, and even in more advanced topics like linear algebra and calculus. Understanding how to apply the Pythagorean theorem to vectors is a crucial skill for anyone working with these mathematical objects. Now that we've calculated the length of ${\vec{p}}$, we've completed all the steps of our problem. Let's take a moment to recap what we've done and appreciate the journey we've taken together.

Conclusion

So, guys, we did it! We successfully found the coordinates and length of vector ${\vec{p}}$. To recap, we followed these steps:

1. Calculated 5${\vec{a}}$: We multiplied each component of ${\vec{a}}$ by 5.
2. Calculated -6${\vec{b}}$: We multiplied each component of ${\vec{b}}$ by -6 (remembering to flip the direction).
3. Calculated ${\vec{p} = 5\vec{a} - 6\vec{b}}$: We added the vectors 5${\vec{a}}$ and -6${\vec{b}}$ component-wise to find the coordinates of ${\vec{p}}$.
4. Calculated the length of ${\vec{p}}$: We used the Pythagorean theorem to find the magnitude of ${\vec{p}}$.

We found that the coordinates of ${\vec{p}}$ are (-5, -1) and its length is ${\sqrt{26}}$. This problem demonstrates the fundamental operations of vector algebra: scalar multiplication and vector addition. By breaking down the problem into smaller steps, we were able to tackle it systematically and arrive at the solution. Remember, practice makes perfect! The more you work with vectors, the more comfortable you'll become with these operations and their applications. So, keep practicing, keep exploring, and keep having fun with math! Vectors are powerful tools, and mastering them will open up a whole new world of possibilities in various fields, from physics and engineering to computer graphics and data science. The ability to manipulate vectors, calculate their magnitudes, and understand their geometric interpretations is a valuable skill that will serve you well in your future endeavors. So, congratulations on completing this problem, and I encourage you to continue your journey of mathematical exploration. And hey, if you enjoyed this breakdown, let me know in the comments! Maybe we can tackle another vector problem together soon. Until then, keep those vectors pointing in the right direction!