Perpendicular Bisector And Collinear Points Vector Problems Explained

Hey guys! Today, we're diving into a couple of cool math problems involving vectors. We'll tackle finding the equation of a perpendicular bisector and then figure out when three points are collinear. Let's get started!

(a) Finding the Vector Equation of the Perpendicular Bisector

This problem is all about lines, midpoints, and perpendicularity in the vector world. We're given two points, A(-3, -9) and B(11, 5), and our mission is to find the vector equation of the perpendicular bisector of the line segment joining these points. Let's break it down step-by-step.

1. Understanding the Perpendicular Bisector

First, let's get clear on what a perpendicular bisector is. It's a line that cuts another line segment into two equal halves and meets it at a right angle (90 degrees). So, we have two key things to find: the midpoint of the line segment and the direction vector of the perpendicular bisector.

2. Finding the Midpoint

The midpoint, let's call it M, is simply the average of the coordinates of the two points. The formula for the midpoint M of a line segment joining points A(x1, y1) and B(x2, y2) is given by:

M = ((x1 + x2)/2, (y1 + y2)/2)

Plugging in our values for A(-3, -9) and B(11, 5), we get:

M = ((-3 + 11)/2, (-9 + 5)/2) = (8/2, -4/2) = (4, -2)

So, the midpoint M is (4, -2). This point will lie on our perpendicular bisector.

3. Finding the Direction Vector of the Line Segment

Next, we need to find the direction vector of the line segment AB. This vector points from A to B and tells us the direction of the line segment. We can find it by subtracting the position vector of A from the position vector of B.

The position vector of a point is simply a vector that starts at the origin and ends at that point. So, the position vector of A is -3i - 9j, and the position vector of B is 11i + 5j.

The direction vector AB is then:

AB = (11i + 5j) - (-3i - 9j) = 11i + 5j + 3i + 9j = 14i + 14j

4. Finding the Direction Vector of the Perpendicular Bisector

Now comes the crucial part: finding the direction vector of the perpendicular bisector. Since the perpendicular bisector is perpendicular to the line segment AB, its direction vector will be orthogonal (at a 90-degree angle) to the direction vector AB. This is where the concept of the negative reciprocal comes into play conceptually, but in vector form, we're looking for a vector whose dot product with AB is zero.

One way to find such a vector is to swap the components of the vector AB and negate one of them. So, if AB is 14i + 14j, a vector perpendicular to it could be -14i + 14j or 14i - 14j. We can simplify either of these by dividing by their greatest common divisor, which is 14, resulting in the direction vectors -i + j or i - j. Let's choose i - j as our direction vector for the perpendicular bisector. It doesn't matter which one we pick, as they both represent the same direction (or opposite directions, which still define the same line).

5. Forming the Vector Equation

We now have everything we need to write the vector equation of the perpendicular bisector. The general form of a vector equation of a line is:

r = a + t * d

where:

• r is the position vector of any point on the line
• a is the position vector of a known point on the line (we'll use the midpoint M)
• t is a scalar parameter (any real number)
• d is the direction vector of the line

We know that a is the position vector of the midpoint M(4, -2), which is 4i - 2j. We also know that d is the direction vector of the perpendicular bisector, which we found to be i - j.

Plugging these into the vector equation, we get:

r = (4i - 2j) + t(i - j)

This is the vector equation of the perpendicular bisector of the line segment joining the points A(-3, -9) and B(11, 5). Awesome! We've successfully navigated the world of vectors and perpendicular bisectors.

(b) Finding the Value of 'n' for Collinear Points

Alright, let's switch gears and tackle the second part of the problem. This time, we're dealing with collinearity. We have three points, P, Q, and R, with position vectors 3i - 2j, ni + j, and 2i - 8j, respectively. Our mission, should we choose to accept it, is to find the value of 'n' that makes these points lie on the same straight line.

1. Understanding Collinearity

So, what does it mean for points to be collinear? Simply put, it means they all lie on the same straight line. Think of it like lining up beads on a string – they all have to be on the same string to be considered collinear. In terms of vectors, this means that the vectors formed by any two pairs of these points must be parallel. This is a crucial concept!

2. Forming Vectors from the Points

Let's form two vectors using the given points. We'll choose vectors PQ and PR. Remember, to find a vector between two points, we subtract the position vector of the starting point from the position vector of the ending point.

• PQ = (ni + j) - (3i - 2j) = (n - 3)i + 3j
• PR = (2i - 8j) - (3i - 2j) = -i - 6j

We now have expressions for the vectors PQ and PR.

3. The Condition for Collinearity: Parallel Vectors

As we discussed earlier, for P, Q, and R to be collinear, vectors PQ and PR must be parallel. What does it mean for vectors to be parallel? It means that one vector is a scalar multiple of the other. In other words, there exists a scalar (let's call it 'k') such that:

PQ = k * PR

This is the key equation we'll use to solve for 'n'.

4. Setting up the Equation and Solving for 'n'

Let's substitute the expressions we found for PQ and PR into the equation:

(n - 3)i + 3j = k * (-i - 6j)

This gives us:

(n - 3)i + 3j = -ki - 6kj

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

1. n - 3 = -k (equating the i components)
2. 3 = -6k (equating the j components)

Let's solve the second equation for 'k':

k = 3 / -6 = -1/2

Now, we can substitute this value of 'k' into the first equation:

n - 3 = -(-1/2) n - 3 = 1/2 n = 1/2 + 3 n = 7/2

So, the value of 'n' that makes the points P, Q, and R collinear is 7/2. Fantastic! We've successfully navigated the conditions for collinearity and solved for the unknown value.

Conclusion

In this mathematical adventure, we conquered two problems involving vectors. We found the vector equation of a perpendicular bisector by understanding midpoints, direction vectors, and perpendicularity. We also determined the value of 'n' that makes three points collinear by leveraging the concept of parallel vectors and scalar multiples. Keep practicing these concepts, and you'll become a vector whiz in no time! You got this! Let me know if you have any questions, guys!