Vector Operations Finding Vectors PQ PR And QR And Position Vector Of Point T

In the realm of three-dimensional geometry, vectors serve as fundamental tools for representing and manipulating spatial relationships. This article delves into vector operations, focusing on finding vectors between points and determining the position vector of a point given certain conditions. We will explore concepts such as vector subtraction, which is crucial for finding the vector pointing from one point to another, and scalar multiplication and vector addition, which are essential for determining the position vector of a point that divides a line segment in a specific ratio. Understanding these operations is vital for solving a wide array of problems in physics, engineering, and computer graphics.

This exploration will be centered around a specific problem involving three points in 3D space, denoted as P, Q, and R, defined by their position vectors with respect to the origin O. The position vectors $\overrightarrow{OP}$, $\overrightarrow{OQ}$, and $\overrightarrow{OR}$ are given as (2, -3, -1), (5, 1, -4), and (3, -3, 2), respectively. Our primary objective is to determine the vectors $\overrightarrow{PQ}$, $\overrightarrow{PR}$, and $\overrightarrow{QR}$, which represent the displacement from point P to Q, P to R, and Q to R. Furthermore, we aim to find the position vector of a point T that lies on the line segment PR and satisfies a given condition related to the ratio of lengths PT and TR. This comprehensive approach will not only provide a solution to the specific problem but also elucidate the underlying principles and methodologies applicable to a broader range of vector-related problems.

To begin, let's delve into the process of finding the vectors $\overrightarrow{PQ}$, $\overrightarrow{PR}$, and $\overrightarrow{QR}$. In vector algebra, the vector pointing from one point to another can be found by subtracting the position vector of the initial point from the position vector of the terminal point. This fundamental principle allows us to express these vectors in terms of the given position vectors $\overrightarrow{OP}$, $\overrightarrow{OQ}$, and $\overrightarrow{OR}$.

The vector $\overrightarrow{PQ}$, representing the displacement from point P to point Q, is calculated by subtracting $\overrightarrow{OP}$ from $\overrightarrow{OQ}$. Mathematically, this is expressed as $\overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP}$. Substituting the given values, we have $\overrightarrow{PQ} = (5, 1, -4) - (2, -3, -1) = (5-2, 1-(-3), -4-(-1)) = (3, 4, -3)$. Thus, the vector $\overrightarrow{PQ}$ is (3, 4, -3). This vector indicates the direction and magnitude of the displacement needed to move from point P to point Q in 3D space.

Similarly, the vector $\overrightarrow{PR}$, representing the displacement from point P to point R, is found by subtracting $\overrightarrow{OP}$ from $\overrightarrow{OR}$. This gives us $\overrightarrow{PR} = \overrightarrow{OR} - \overrightarrow{OP} = (3, -3, 2) - (2, -3, -1) = (3-2, -3-(-3), 2-(-1)) = (1, 0, 3)$. The vector $\overrightarrow{PR}$ is therefore (1, 0, 3). This vector describes the displacement required to move from point P to point R.

Lastly, the vector $\overrightarrow{QR}$, representing the displacement from point Q to point R, is calculated by subtracting $\overrightarrow{OQ}$ from $\overrightarrow{OR}$. This yields $\overrightarrow{QR} = \overrightarrow{OR} - \overrightarrow{OQ} = (3, -3, 2) - (5, 1, -4) = (3-5, -3-1, 2-(-4)) = (-2, -4, 6)$. Hence, the vector $\overrightarrow{QR}$ is (-2, -4, 6). This vector represents the displacement from point Q to point R.

In summary, we have determined the vectors $\overrightarrow{PQ}$ as (3, 4, -3), $\overrightarrow{PR}$ as (1, 0, 3), and $\overrightarrow{QR}$ as (-2, -4, 6). These vectors provide a comprehensive description of the spatial relationships between the points P, Q, and R, forming the foundation for further analysis and problem-solving in three-dimensional geometry. The ability to accurately calculate these vectors is a crucial skill in various fields, including computer graphics, physics simulations, and engineering design.

Now, let's turn our attention to determining the position vector of point T. We are given that point T lies on the line segment PR and that PT = (1/2)TR. This condition provides a crucial relationship between the lengths of the segments PT and TR, which we can leverage to find the position vector of T. To tackle this, we'll utilize the concept of dividing a line segment in a given ratio using vector algebra.

The condition PT = (1/2)TR implies that the ratio of PT to TR is 1:2. This means that point T divides the line segment PR internally in the ratio 1:2. We can use the section formula for vectors to find the position vector of T. The section formula states that if a point divides a line segment joining points A and B with position vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ in the ratio m:n, then the position vector of the point is given by $\overrightarrow{OT} = (n\overrightarrow{OA} + m\overrightarrow{OB}) / (m + n)$.

In our case, point T divides the line segment PR in the ratio 1:2. Therefore, m = 1 and n = 2. The position vectors of P and R are $\overrightarrow{OP} = (2, -3, -1)$ and $\overrightarrow{OR} = (3, -3, 2)$, respectively. Applying the section formula, we get the position vector of T as:

$\overrightarrow{OT} = (2\overrightarrow{OP} + 1\overrightarrow{OR}) / (1 + 2)$ $\overrightarrow{OT} = (2(2, -3, -1) + 1(3, -3, 2)) / 3$ $\overrightarrow{OT} = ((4, -6, -2) + (3, -3, 2)) / 3$ $\overrightarrow{OT} = (7, -9, 0) / 3$ $\overrightarrow{OT} = (7/3, -3, 0)$

Thus, the position vector of point T is (7/3, -3, 0). This vector pinpoints the location of T in 3D space, satisfying the given condition that it lies on the line segment PR and divides it in the ratio 1:2. The section formula provides a powerful tool for solving problems involving division of line segments in vector algebra, and its application here demonstrates its utility in determining the position of a point based on its relative location on a line segment.

In conclusion, we have successfully navigated the problem of vector operations and point determination in 3D space. We began by finding the vectors $\overrightarrow{PQ}$, $\overrightarrow{PR}$, and $\overrightarrow{QR}$ using vector subtraction, which allowed us to describe the displacement between points P, Q, and R. These vectors serve as essential tools for understanding the spatial relationships between these points.

Furthermore, we determined the position vector of point T, which lies on the line segment PR and satisfies the condition PT = (1/2)TR. This was accomplished by applying the section formula, a fundamental concept in vector algebra that enables us to find the position vector of a point dividing a line segment in a given ratio. The resulting position vector $\overrightarrow{OT} = (7/3, -3, 0)$ provides a precise location for point T in 3D space.

This exploration has highlighted the power and versatility of vector operations in solving geometric problems. Vector subtraction, scalar multiplication, vector addition, and the section formula are invaluable tools for manipulating and analyzing spatial relationships. The concepts and methodologies discussed here have broad applications in various fields, including physics, engineering, computer graphics, and more. A solid understanding of these principles is crucial for anyone working with spatial data and geometric problems.

By mastering these techniques, one can effectively tackle a wide range of problems involving points, lines, and vectors in three-dimensional space. The ability to accurately perform vector operations and determine the position of points based on given conditions is a cornerstone of advanced mathematical and scientific problem-solving.

Vectors, Position Vectors, Vector Subtraction, Scalar Multiplication, Section Formula, 3D Geometry, Displacement Vectors, Ratio of Lengths, Line Segment Division, Point Determination