Vector PQ Magnitude & Section Formula Calculation

Hey guys! Today, we're diving into a cool problem involving vectors. We'll tackle how to find the magnitude of a vector formed by two points and then figure out the position vector of a point that divides a line segment in a given ratio. It might sound a bit complex, but trust me, we'll break it down step-by-step so it's super easy to understand. So, grab your thinking caps, and let's get started!

Understanding Position Vectors and Vector PQ

First, let's clarify what position vectors actually are. Imagine a coordinate system; a position vector is simply a vector that points from the origin (the heart of our coordinate system) to a specific point. It tells us exactly where that point is located in space. In our problem, we're given two points, P and Q, each with its own position vector. The position vector p is represented as 2i + 3k, and the position vector q is 3i + 4j. Here, i, j, and k are the unit vectors along the x, y, and z axes, respectively. Think of them as the fundamental directions in our 3D world.

Now, what about the vector PQ? Well, this vector represents the displacement from point P to point Q. It's like drawing an arrow starting at P and ending at Q. To find PQ, we use a neat trick: we subtract the position vector of the starting point (P) from the position vector of the ending point (Q). Mathematically, this looks like PQ = q - p. Let's plug in the values we have: PQ = (3i + 4j) - (2i + 3k). Simplifying this, we get PQ = (3i - 2i) + 4j - 3k, which boils down to PQ = i + 4j - 3k. This is the vector that points directly from P to Q, telling us how to move from P to reach Q.

To summarize, we've taken the given position vectors, understood what they represent, and then used them to find the vector PQ. This is a crucial first step because the magnitude of this vector will tell us the distance between the two points, which is exactly what part (a) of our problem asks for. Understanding this foundation makes the rest of the problem much smoother. So, before we move on, make sure you're comfortable with how we found PQ – it's the key to unlocking the rest of the solution!

Calculating the Magnitude of Vector PQ

Alright, now that we've successfully found the vector PQ, which is i + 4j - 3k, let's move on to the next part: finding its magnitude. The magnitude of a vector is essentially its length – it tells us how long the arrow representing the vector actually is. In our case, it represents the distance between points P and Q. Calculating the magnitude involves a bit of mathematical magic using the components of the vector.

The formula to find the magnitude of a vector is actually quite straightforward. If we have a vector, let's say v = ai + bj + ck, then the magnitude of v, denoted as |v|, is given by the square root of the sum of the squares of its components: |v| = √(a² + b² + c²). It might look intimidating, but it's really just applying the Pythagorean theorem in three dimensions! We're essentially finding the hypotenuse of a right triangle in 3D space.

Let's apply this to our vector PQ = i + 4j - 3k. Here, the components are a = 1 (the coefficient of i), b = 4 (the coefficient of j), and c = -3 (the coefficient of k). Plugging these into our formula, we get |PQ| = √(1² + 4² + (-3)²). Let's simplify this step by step. First, we square each component: 1² = 1, 4² = 16, and (-3)² = 9. Then, we add these squared values together: 1 + 16 + 9 = 26. Finally, we take the square root of the sum: |PQ| = √26. Now, the problem asks for the answer to four significant figures. Using a calculator, we find that √26 is approximately 5.09901951359... Rounding this to four significant figures gives us 5.099. So, the magnitude of vector PQ, or the distance between points P and Q, is approximately 5.099 units.

In summary, we used the formula for the magnitude of a vector and applied it to PQ. We squared the components, summed them up, took the square root, and then rounded the result to the required precision. This process might seem a bit technical, but it's a fundamental skill in vector mathematics. We've now successfully calculated the distance between points P and Q, answering the first part of our problem. Let's take a deep breath and move on to the next part, where we'll explore the concept of dividing a line segment in a given ratio.

Finding the Position Vector of Point T Using the Section Formula

Now, let's tackle the second part of our problem, which involves finding the position vector of a point T that divides the line segment PQ in the ratio 2:3. This is where the section formula comes into play. The section formula is a handy tool that helps us determine the position vector of a point that divides a line segment joining two other points in a specific ratio. It's like having a recipe for finding the exact location of a point along a line.

The section formula states that if a point T divides the line segment joining points P and Q in the ratio m:n, then the position vector of T, denoted as t, can be calculated as: t = (np + mq) / (m + n). Don't let the formula intimidate you; it's actually quite logical. We're essentially taking a weighted average of the position vectors p and q, where the weights are determined by the ratio m:n. The ratio tells us how far along the line segment T is located, relative to P and Q.

In our problem, we know that T divides PQ in the ratio 2:3. This means m = 2 and n = 3. We also know the position vectors of P and Q: p = 2i + 3k and q = 3i + 4j. Now, it's just a matter of plugging these values into the section formula. Let's do it step by step: t = (3(2i + 3k) + 2(3i + 4j)) / (2 + 3). First, we distribute the scalars: 3(2i + 3k) = 6i + 9k and 2(3i + 4j) = 6i + 8j. Now, we add these vectors together: (6i + 9k) + (6i + 8j) = 12i + 8j + 9k. Finally, we divide by the sum of the ratio, which is 2 + 3 = 5: t = (12i + 8j + 9k) / 5. We can rewrite this as t = (12/5)i + (8/5)j + (9/5)k. This is the position vector of point T.

So, to recap, we used the section formula, plugged in the given ratio and position vectors, and simplified the expression to find the position vector of T. This formula is a powerful tool for solving problems involving division of line segments, and it's a great addition to your vector toolkit. We've now successfully found the position vector of point T, completing the second part of our problem. Give yourselves a pat on the back – we're making great progress!

Wrapping Up: Putting It All Together

Alright guys, we've reached the end of our vector adventure! Let's take a moment to recap what we've accomplished. We started with a problem that gave us the position vectors of two points, P and Q, and asked us to find two things: the magnitude of vector PQ and the position vector of a point T that divides the line segment PQ in the ratio 2:3.

First, we tackled the magnitude of vector PQ. We understood that this magnitude represents the distance between points P and Q. To find it, we first calculated the vector PQ by subtracting the position vector of P from the position vector of Q. Then, we used the magnitude formula, which involves squaring the components of the vector, summing them up, and taking the square root. We carefully rounded our final answer to four significant figures, as requested in the problem.

Next, we moved on to finding the position vector of point T. This is where the section formula came to our rescue. We learned that the section formula allows us to calculate the position vector of a point that divides a line segment in a given ratio. We plugged in the given ratio and position vectors into the formula, carefully simplified the expression, and arrived at the position vector of T.

Throughout this problem, we've not only applied mathematical formulas but also gained a deeper understanding of what these formulas represent. We've visualized position vectors, understood the concept of vector magnitude as distance, and used the section formula as a tool for dividing line segments. These are fundamental concepts in vector mathematics, and mastering them will open doors to more advanced topics in physics, engineering, and computer graphics.

So, what's the key takeaway here? It's that breaking down complex problems into smaller, manageable steps is crucial. We started with a seemingly intimidating problem, but by understanding the underlying concepts, applying the right formulas, and working step by step, we were able to solve it successfully. Keep practicing, keep exploring, and keep asking questions. Math is a journey, and every problem we solve is a step forward. Until next time, keep those vectors pointing in the right direction!