Vector OR: Finding It In Terms Of Vectors P And Q

Hey guys! Let's dive into a cool vector problem today. We're given two vectors, ${\vec{OP} = \vec{p}}$ and ${\vec{OQ} = \vec{q}}$, and a point R that divides the line segment PQ in a specific ratio. Our mission? To express ${\vec{OR}}$ in terms of ${\vec{p}}$ and ${\vec{q}}$. Sounds like fun, right? Let's break it down step by step.

Understanding the Problem

First, let’s make sure we fully grasp what the question is asking. We've got points P and Q, defined by their position vectors ${\vec{p}}$ and ${\vec{q}}$ respectively, relative to the origin O. Point R lies somewhere on the line segment PQ. The crucial piece of information is that 2PR = PQ. This tells us how R divides the segment PQ. We need to find the position vector ${\vec{OR}}$, which points from the origin to point R, but we want to express it using only ${\vec{p}}$ and ${\vec{q}}$. This type of problem is common in vector algebra, and it's a great way to flex our understanding of vector operations and ratios. So, grab your thinking caps, and let’s get started!

Visualizing the Scenario

Before we jump into calculations, it's super helpful to visualize what’s going on. Imagine a coordinate plane with the origin O. Points P and Q are positioned somewhere in this plane, defined by their vectors ${\vec{p}}$ and ${\vec{q}}$. Now, draw a line segment connecting P and Q. Point R lies on this line segment. The condition 2PR = PQ means that the distance from P to R is half the distance from P to Q. In other words, R is the midpoint of the segment PQ. This visual representation gives us a clearer picture and helps us anticipate the relationships between the vectors. When we break down complex problems like this visually, it becomes easier to identify the steps we need to take to reach the solution. Vector problems often benefit from a quick sketch to solidify the understanding of spatial relationships.

Key Concepts: Position Vectors and Section Formula

To tackle this problem, we need to remember a couple of key concepts from vector algebra. First, position vectors are vectors that describe the position of a point relative to the origin. In our case, ${\vec{OP}}$, ${\vec{OQ}}$, and ${\vec{OR}}$ are all position vectors. They tell us exactly where points P, Q, and R are located in space relative to O. Second, we need to recall the section formula. This formula is incredibly useful when dealing with points that divide a line segment in a given ratio. It allows us to express the position vector of the dividing point in terms of the position vectors of the endpoints and the ratio of division. Understanding and applying the section formula is crucial for solving this type of problem efficiently. Make sure you're comfortable with these concepts before moving on!

Applying the Section Formula

Okay, now for the exciting part – using the section formula to find ${\vec{OR}}$. The section formula states that if a point R divides the line segment PQ in the ratio m:n, then the position vector of R is given by:

${\vec{OR} = \frac{n\vec{p} + m\vec{q}}{m + n}}$

But wait! We're given 2PR = PQ, not a direct ratio. We need to translate this information into a ratio. If 2PR = PQ, then PR = (1/2)PQ. This means that PR : RQ is 1 : 1, because if PR is half of PQ, then RQ must be the other half, making them equal. Therefore, R is the midpoint of PQ.

Determining the Ratio

Let's make sure we've got this ratio thing nailed down. We know that 2PR = PQ. This tells us something important about the relationship between the lengths of the segments PR and PQ. To figure out the ratio in which R divides PQ, we need to think about how the segment lengths relate to each other. Since PR is half of PQ, the remaining part of the segment, RQ, must also be equal to PR. This means PR and RQ are equal in length. Therefore, R divides PQ into two equal parts. This directly translates to a ratio of 1:1. So, when we use the section formula, we'll plug in m = 1 and n = 1. Getting this ratio right is crucial because it's the foundation for the rest of our calculation. Double-checking this step can save you from making mistakes later on.

Plugging into the Formula

Now we have all the pieces of the puzzle! We know the section formula, and we've figured out the ratio in which R divides PQ (which is 1:1). It's time to plug the values into the formula. Remember, the section formula is:

${\vec{OR} = \frac{n\vec{p} + m\vec{q}}{m + n}}$

In our case, m = 1 and n = 1. So, we substitute these values, along with ${\vec{OP} = \vec{p}}$ and ${\vec{OQ} = \vec{q}}$, into the formula:

${\vec{OR} = \frac{1*\vec{p} + 1*\vec{q}}{1 + 1}}$

This simplifies to:

${\vec{OR} = \frac{\vec{p} + \vec{q}}{2}}$

And there you have it! We've successfully expressed ${\vec{OR}}$ in terms of ${\vec{p}}$ and ${\vec{q}}$. This result shows that ${\vec{OR}}$ is simply the average of the vectors ${\vec{p}}$ and ${\vec{q}}$, which makes sense since R is the midpoint of PQ.

The Solution: ${\vec{OR} = \frac{\vec{p} + \vec{q}}{2}}$

Woohoo! We've made it to the solution. After carefully applying the section formula and understanding the given ratio, we found that:

${\vec{OR} = \frac{\vec{p} + \vec{q}}{2}}$

This means the position vector ${\vec{OR}}$ is the average of the position vectors ${\vec{p}}$ and ${\vec{q}}$. This result aligns perfectly with our understanding that R is the midpoint of the line segment PQ. When a point is the midpoint, its position vector is simply the average of the position vectors of the endpoints. This problem highlights the power of the section formula and how it allows us to relate vectors and ratios in geometric problems. It's a fundamental concept in vector algebra, and mastering it will definitely help you tackle similar challenges in the future. So, give yourself a pat on the back for working through this problem with me!

Interpretation of the Result

Let’s take a moment to really understand what our solution means. The equation ${\vec{OR} = \frac{\vec{p} + \vec{q}}{2}}$ tells us that the vector ${\vec{OR}}$ is the result of adding the vectors ${\vec{p}}$ and ${\vec{q}}$ together and then dividing by 2. Geometrically, this means we're finding the midpoint of the line segment connecting the points P and Q. Think about it: adding ${\vec{p}}$ and ${\vec{q}}$ gives us a vector that represents the diagonal of the parallelogram formed by ${\vec{p}}$ and ${\vec{q}}$. Dividing by 2 effectively scales this diagonal vector down to reach the midpoint. This interpretation gives us a deeper insight into the relationship between vector operations and geometric concepts. It reinforces the idea that vector algebra provides a powerful tool for describing and analyzing geometric figures. Understanding the meaning behind the math is just as important as getting the right answer!

Conclusion

So, there you have it! We successfully found ${\vec{OR}}$ in terms of ${\vec{p}}$ and ${\vec{q}}$ by using the section formula and a bit of geometric reasoning. Remember, the key to solving these types of problems is to carefully understand the given information, visualize the scenario, and apply the appropriate formulas. Vector algebra can seem tricky at first, but with practice and a solid grasp of the fundamental concepts, you'll be solving complex problems like a pro in no time. Keep practicing, keep exploring, and most importantly, keep having fun with math! You guys nailed it!