Question

Let \(\mathbf{p}\) and \(\mathbf{q}\) be the vectors of points \(P\) and \(Q\), respectively, and let \(R\) be the point whose vector is \(\mathbf{p}+\mathbf{q}\). Express the following in terms of \(\mathbf{p}\) and \(\mathbf{q}\). a. \(\overrightarrow{Q P}\) b. \(\overrightarrow{Q R}\) c. \(\overrightarrow{R P}\) d. \(\overrightarrow{R O}\) where \(O\) is the origin

Short answer

a. \( \mathbf{p} - \mathbf{q} \); b. \( \mathbf{p} \); c. \(-\mathbf{q}\); d. \(-\mathbf{p} - \mathbf{q} \).

Step-by-step solution

Understand Vector Representation

A vector is represented by its initial and terminal points. For example, vector \( \overrightarrow{PQ} \) is formed by finding the difference between the position vectors of points \( Q \) and \( P \). This can be written as the vector \( \mathbf{q} - \mathbf{p} \). Similarly, for any directed line segment from point \( A \) to point \( B \), the vector is represented as \( \overrightarrow{BA} = \mathbf{a} - \mathbf{b} \).

Calculate \( \overrightarrow{Q P} \)

To find the vector \( \overrightarrow{Q P} \), we start from \( Q \) and move towards \( P \). Using the formula \( \overrightarrow{BA} = \mathbf{a} - \mathbf{b} \), we have: \[ \overrightarrow{Q P} = \mathbf{p} - \mathbf{q} \] Thus, \( \overrightarrow{Q P} \) is expressed as \( \mathbf{p} - \mathbf{q} \).

Calculate \( \overrightarrow{Q R} \)

For \( \overrightarrow{Q R} \), we start from \( Q \) and end at \( R \). The vector \( R \) is given by \( \mathbf{p} + \mathbf{q} \). Therefore, we have: \[ \overrightarrow{Q R} = (\mathbf{p} + \mathbf{q}) - \mathbf{q} = \mathbf{p} \] So, \( \overrightarrow{Q R} = \mathbf{p} \).

Calculate \( \overrightarrow{R P} \)

For \( \overrightarrow{R P} \), we start from \( R \) and move towards \( P \). We have: \[ \overrightarrow{R P} = \mathbf{p} - (\mathbf{p} + \mathbf{q}) = \mathbf{p} - \mathbf{p} - \mathbf{q} = -\mathbf{q} \] Hence, \( \overrightarrow{R P} = -\mathbf{q} \).

Calculate \( \overrightarrow{R O} \)

To find \( \overrightarrow{R O} \), we recognize that \( O \) is the origin represented by the vector \( \mathbf{0} \). Starting from \( R \) and going to \( O \), we have: \[ \overrightarrow{R O} = \mathbf{0} - (\mathbf{p} + \mathbf{q}) = -\mathbf{p} - \mathbf{q} \] So, \( \overrightarrow{R O} = -\mathbf{p} - \mathbf{q} \).

Key concepts

Point Representation in Vectors

Vectors are an essential part of understanding geometry and physics. In vector arithmetic, one key concept is the representation of points in terms of vectors. Each point in space can be associated with a vector, often called its position vector.
This association treats the origin of the coordinate system as the reference point. For example, if point \( P \) is represented by vector \( \mathbf{p} \), it means that \( \mathbf{p} \) specifies the position of \( P \) relative to the origin. Similarly, vector \( \mathbf{q} \) can be used to represent point \( Q \).
By knowing the position vectors of two points, mathematical operations such as addition and subtraction can help describe the relationship between those points in space.

Vector Subtraction

When working with vectors, subtraction is a critical operation. This operation helps determine the vector between two points in space.
For instance, to find the vector from point \( Q \) to point \( P \), represented as \( \overrightarrow{QP} \), we perform vector subtraction: \( \mathbf{p} - \mathbf{q} \). This operation effectively "leads" us from point \( Q \) towards point \( P \).
The same principle applies in a general scenario: for any points \( A \) and \( B \), the vector \( \overrightarrow{BA} \) can be found using \( \mathbf{a} - \mathbf{b} \). This subtraction of vectors helps us navigate through space, providing direction and magnitude from one point to another.

Position Vectors

Position vectors play a crucial role in depicting the location of points within vector space. They represent a point's position concerning the origin.
Consider vector \( \mathbf{p} \) representing point \( P \) and vector \( \mathbf{q} \) representing point \( Q \). By combining these vectors, we can form new vectors, such as \( \mathbf{p} + \mathbf{q} \), which represents a point \( R \) located at the vector sum of positions \( \mathbf{p} \) and \( \mathbf{q} \).
This method of combining vectors provides a flexible way to express points and their connections in the spatial context. It extends further, enabling us to construct vectors like \( \overrightarrow{QR} \), where \( R \) is the vector addition of \( P \) and \( Q \).

Origin in Vector Space

Understanding the origin in vector space is critical when dealing with position vectors and operations involving them. The origin is typically denoted by the vector \( \mathbf{0} \). It serves as the point of reference for all position vectors.
In exercises involving vector expressions, such as calculating \( \overrightarrow{RO} \) where \( O \) is the origin, it simplifies to subtracting \( R \)'s vector from the zero vector, which is essentially negating the position vector of \( R \).
This foundational point allows for the simplification of various vector operations, setting the stage for more complex navigations in space, like finding the vector leading back to the origin which gives you a sense of the reverse direction of a position vector.