Problem 10 - Contemporary Precalculus

Question

Find a vector with the origin as initial point that is equivalent to the vector \(\overrightarrow{P Q}\). $$P=(\sqrt{2}, 4), Q=(\sqrt{3},-1)$$

Step-by-Step Solution

Answer

Question: Given points P(√2, 4) and Q(√3, -1), find a vector with the origin as the initial point that is equivalent to the vector from P to Q. Answer: The vector equivalent to the vector from P to Q with the origin as the initial point is (√3 - √2, -5).

Step 1: Finding the Vector from P to Q

To find the vector from P to Q, subtract the coordinates of point P from the coordinates of point Q:$$\overrightarrow{PQ} = (Q_x - P_x, Q_y - P_y)$$Plugging in the given coordinates for P and Q, we get:$$\overrightarrow{PQ} = (\sqrt{3} - \sqrt{2}, -1 - 4)$$

Step 2: Simplifying the Vector

Now we can simplify the vector:$$\overrightarrow{PQ} = (\sqrt{3} - \sqrt{2}, -5)$$

Step 3: Translating the Vector to Start at the Origin

Since we want a vector that starts at the origin and has the same magnitude and direction as \(\overrightarrow{P Q}\), we can simply use the vector that we found in step 2. The vector equivalent to \(\overrightarrow{P Q}\) with the origin as the initial point is:$$\overrightarrow{OP} = (\sqrt{3} - \sqrt{2}, -5)$$

Key Concepts

Vectors

Vectors are mathematical entities that have both a direction and a magnitude. You can think of them as arrows that point in a specific direction with a certain length, which represents their magnitude.
They are used widely in both physics and engineering, but also in computer graphics and many other fields.
A vector is often represented in a coordinate system as a pair or set of numbers.

• The first component often corresponds to the horizontal direction or x-axis.
• The second component corresponds to the vertical direction or y-axis.

In the given exercise, we have a vector from point P to point Q, represented as \(\overrightarrow{PQ}\). This vector describes how to move from point P to point Q using these directions.
Knowing how to work with vectors is fundamental to understanding many concepts in mathematics.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic principles to solve geometric problems.
It uses a coordinate system to describe geometric figures and their properties.
By plotting points on a x-y plane, we can visualize and compute the relationships between them.In the problem, we use the Cartesian coordinate system where each point is described by a pair of coordinates \(x, y\).
Point P has the coordinates \((\sqrt{2}, 4)\) and point Q the coordinates \((\sqrt{3}, -1)\).

• The x-coordinate represents the horizontal position.
• The y-coordinate represents the vertical position.

Coordinate geometry helps us understand how these points are positioned relative to each other and allows us to calculate the vector \(\overrightarrow{PQ}\), by subtracting their corresponding coordinates. This is a useful method for translating geometric problems into easier solvable algebraic equations.

Vector Subtraction

Vector subtraction is a key operation in vector mathematics. It involves finding the difference between two vectors, often to determine the direction and magnitude from one point to another.
In coordinate geometry, subtracting the coordinates of one point from another results in a new vector.
This new vector effectively describes the translation from the first point to the second.In the original exercise, we subtract the coordinates of point P from those of point Q to find the vector \(\overrightarrow{PQ}\):

• This is done by calculating \(Q_x - P_x\) for the x-coordinate.
• And \(Q_y - P_y\) for the y-coordinate.

Once these operations are completed, we have the vector \((\sqrt{3} - \sqrt{2}, -5)\).
This vector is equivalent to the vector from the origin with the same direction and magnitude—a crucial aspect when translating vectors. Understanding vector subtraction is fundamental for manipulating and analyzing vectors in various applications of mathematics.