Question

Point P has coordinates \((7,8)\). Point \(Q\) has coordinates \((-2,4)\) (a) Draw a sketch showing \(\mathrm{P}\) and \(\mathrm{Q}\). (b) State the position vectors of \(\mathrm{P}\) and \(\mathrm{Q}\). (c) Find an expression for \(\overrightarrow{P Q}\). (d) Find \(|\overrightarrow{\mathrm{PQ}}|\).

Short answer

Answer: The expression for the vector from point P to point Q is \(\overrightarrow{PQ} = \begin{pmatrix} -9 \\ -4 \end{pmatrix}\), and its magnitude is \(\sqrt{97}\).

Step-by-step solution

(a) Drawing a Sketch of P and Q)

To draw a sketch of points P and Q, plot the given coordinates (\((7,8)\) for point P and \((-2,4)\) for point Q) on the Cartesian plane.

(b) Position Vectors of P and Q)

To find the position vectors of points P and Q, write the given coordinates as column vectors: Position vector of P: \(\begin{pmatrix} 7 \\ 8 \end{pmatrix}\) Position vector of Q: \(\begin{pmatrix} -2 \\ 4 \end{pmatrix}\)

(c) Expression for \(\overrightarrow{PQ}\))

To find the expression for \(\overrightarrow{PQ}\), subtract the position vector of P from the position vector of Q, that is: \(\overrightarrow{PQ} = \begin{pmatrix} -2 \\ 4 \end{pmatrix} - \begin{pmatrix} 7 \\ 8 \end{pmatrix}\) Now perform the subtraction: \(\overrightarrow{PQ} = \begin{pmatrix} -2 - 7 \\ 4 - 8 \end{pmatrix} = \begin{pmatrix} -9 \\ -4 \end{pmatrix}\)

(d) Magnitude of \(\overrightarrow{PQ}\))

To find the magnitude of \(\overrightarrow{PQ}\), use the magnitude formula for a 2-dimensional vector: \(|\overrightarrow{PQ}| = \sqrt{(-9)^2 + (-4)^2}\) Calculate the magnitude: \(|\overrightarrow{PQ}| = \sqrt{81 + 16} = \sqrt{97}\) So, the magnitude of \(\overrightarrow{PQ}\) is \(\sqrt{97}\).

Key concepts

Position Vector

In vector algebra, a position vector is a crucial concept that helps identify the location of a point in space. To find the position vector of a point, we take its coordinates and express them as a vector that originates from the origin of our coordinate system. For example, point P with coordinates \((7,8)\) has a position vector represented as \(\begin{pmatrix} 7 \ 8 \end{pmatrix}\). The same approach applies to any other point, such as point Q with coordinates \((-2,4)\), which translates to the position vector \(\begin{pmatrix} -2 \ 4 \end{pmatrix}\).

These position vectors allow us to analyze points within the scope of vector algebra, and they serve as the backbone for operations like vector subtraction and finding vector magnitudes. They map the points directly to the vector space, giving a clear, visual representation of their positions.

Magnitude of a Vector

The magnitude of a vector, also known as its length, measures how long the vector is. Imagine it as the straight-line distance from the initial point to the terminal point. For any vector \ \(\overrightarrow{AB} = \begin{pmatrix} x \ y \end{pmatrix} \ \), we calculate its magnitude using the formula:

\[ |\overrightarrow{AB}| = \sqrt{x^2 + y^2} \]

In the given exercise, after calculating the expression \(\overrightarrow{PQ} = \begin{pmatrix} -9 \ -4 \end{pmatrix}\), the magnitude is found with:
\[ |\overrightarrow{PQ}| = \sqrt{(-9)^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97} \]
This result, \(\sqrt{97}\), represents the magnitude of the vector between points P and Q. Understanding the magnitude is essential for comprehending the size of a vector, independent of its direction.

Vector Subtraction

Vector subtraction is a fundamental operation in vector algebra that allows us to find a vector representing the difference between two points. The operation involves subtracting the corresponding components of two vectors. To subtract vector \(\mathbf{b}\) from vector \(\mathbf{a}\), represented as \(\overrightarrow{AB}\), we use:

\[ \overrightarrow{AB} = \begin{pmatrix} a_1 \ a_2 \end{pmatrix} - \begin{pmatrix} b_1 \ b_2 \end{pmatrix} \]

For our exercise, we determine \(\overrightarrow{PQ}\) as follows:
start with the position vectors \(\overrightarrow{P} = \begin{pmatrix} 7 \ 8 \end{pmatrix}\) and \(\overrightarrow{Q} = \begin{pmatrix} -2 \ 4 \end{pmatrix}\)

Then, \(\overrightarrow{PQ} = \overrightarrow{Q} - \overrightarrow{P}\):
\[ \begin{pmatrix} -2 \ 4 \end{pmatrix} - \begin{pmatrix} 7 \ 8 \end{pmatrix} = \begin{pmatrix} -9 \ -4 \end{pmatrix} \]

This new vector \(\begin{pmatrix} -9 \ -4 \end{pmatrix}\) depicts the direction and distance from point P to point Q, assisting in vector geometry applications and interpretations.