Question

Define the points \(P(-4,1), Q(3,-4),\) and \(R(2,6) .\) Carry out the following calculations. Express \(\overrightarrow{P Q}\) in the form \(a \mathbf{i}+b \mathbf{j}\).

Short answer

Question: Find the vector \(\overrightarrow{P Q}\) when point P has coordinates (-4, 1) and point Q has coordinates (3, -4). Express the vector in the form \(a \mathbf{i} + b \mathbf{j}\). Answer: The vector \(\overrightarrow{P Q}\) is calculated as \(7 \mathbf{i} - 5 \mathbf{j}\).

Step-by-step solution

Identify the coordinates of points P and Q

We have point P with coordinates \((-4,1)\) and point Q with coordinates \((3,-4)\).

Calculate \(\overrightarrow{P Q}\)

Subtract the coordinates of point P from the coordinates of point Q to find the vector \(\overrightarrow{P Q}\), like this: \(\overrightarrow{P Q} = (3-(-4), -4-1)\)

Simplify the coordinates

Simplify the coordinates of vector \(\overrightarrow{P Q}\): \(\overrightarrow{P Q} = (3+4, -4-1) = (7, -5)\)

Express the vector in \(a \mathbf{i} + b \mathbf{j}\) form

We have the vector \(\overrightarrow{P Q} = (7, -5)\). We can express it in the form \(a \mathbf{i} + b \mathbf{j}\) as: \(\overrightarrow{P Q} = 7 \mathbf{i} - 5 \mathbf{j}\)

Key concepts

Vector Addition

Vector addition is a fundamental concept in vector math, involving the combination of two or more vectors to yield a resultant vector. It's like adding arrows that have both direction and magnitude. In the context of our problem, we consider vectors as directions from one point to another.To add two vectors, follow these simple steps:

• Break down each vector into its components along the x-axis and y-axis.
• For vectors represented in coordinate form, it’s as simple as adding corresponding components together. For example, if you have vectors \( \overrightarrow{A} = (x_1, y_1) \) and \( \overrightarrow{B} = (x_2, y_2) \), the sum \( \overrightarrow{A} + \overrightarrow{B} \) is \((x_1 + x_2, y_1 + y_2)\).

In our exercise, we only calculated a single vector \( \overrightarrow{PQ} \), but understanding vector addition is essential when dealing with multiple vectors. This concept becomes particularly helpful when solving problems involving conditions like balancing forces or moving from one position to another in space.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry using a coordinate system. In our given exercise, we utilize coordinate geometry to define and calculate vectors like \( \overrightarrow{PQ} \).The process generally involves:

• Assigning each point on a plane a unique coordinate pair (x, y).
• Using these coordinates to calculate distances and directions between points, which can be expressed as vectors.

Here, the points \( P(-4,1) \) and \( Q(3,-4) \) are used to determine the vector \( \overrightarrow{PQ} \). The coordinates give us a straightforward method to compute this vector by performing basic arithmetic operations between corresponding coordinates of the two points.This approach is powerful, enabling the examination of geometrical properties and configurations purely through algebraic computations. It provides a link between abstract geometrical concepts and concrete algebraic expressions.

Vector Notation

Vector notation is a systematic way to express vectors and their properties. In mathematical texts and problems, it's important to accurately notate vectors to avoid confusion and ensure clear communication.When addressing vectors in a coordinate system:

• Vectors are often expressed in the form \( a \mathbf{i} + b \mathbf{j} \), where \( a \) and \( b \) are the coefficients that represent the vector's components in the direction of the x-axis and y-axis, respectively.
• In our example, the vector \( \overrightarrow{PQ} = (7, -5) \) translates to \( 7 \mathbf{i} - 5 \mathbf{j} \). This notation clearly describes the direction and magnitude of the vector.

This kind of notation is concise and makes it easier to grasp the concept of vectors as multi-dimensional quantities. With practice, vector notation becomes second nature, simplifying calculations and enhancing the understanding of vector operations in various mathematical and scientific applications.