Question

For the points \(P=(-4,3)\) and \(Q=(5,-1)\) write the vector \(\mathbf{v}\) represented by the directed line segment \(\overrightarrow{P Q}\) in the form \(a \mathbf{i}+b \mathbf{j}\) and find \(\|\mathbf{v}\|\).

Short answer

The vector is \(\mathbf{v} = 9 \mathbf{i} - 4 \mathbf{j} \) and its magnitude is \ \sqrt{97} \.

Step-by-step solution

- Define the Points

Identify the coordinates of point P as \(P = (-4, 3)\) and the coordinates of point Q as \(Q = (5, -1)\).

- Find the Components of the Vector

The vector \(\mathbf{v}\) represented by the directed line segment \(\overrightarrow{PQ}\) can be found by subtracting the coordinates of P from the coordinates of Q. Calculate the components \(a\) and \(b\) as follows: \ a = 5 - (-4) = 5 + 4 = 9 \, \ b = -1 - 3 = -4 \.

- Express the Vector

Write the vector \(\mathbf{v}\) in the form \(a \mathbf{i} + b \mathbf{j} \). Therefore, the vector \(\mathbf{v}\) is \(\mathbf{v} = 9 \mathbf{i} - 4 \mathbf{j} \).

- Calculate the Magnitude of the Vector

To find the magnitude \(\|\mathbf{v}\|\), use the formula \ \|\mathbf{v}\| = \sqrt{a^2 + b^2} \. Here, \ a = 9 \ and \ b = -4\. Plug in these values: \ \|\mathbf{v}\| = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97} \.

Key concepts

vector representation

A vector represents a quantity that has both magnitude and direction. In the context of coordinate geometry, it can be visualized as an arrow pointing from one point to another.
For example, the vector \mathbf{v}\ is represented by the directed line segment \overrightarrow{PQ}\, connecting points \(P = (-4, 3)\) and \(Q = (5, -1)\). This vector is determined by the difference in coordinates between the points.
When presenting a vector in standard form, we often use \mathbf{i}\ and \mathbf{j}\ to represent unit vectors along the x and y axes, respectively. Hence, the vector \mathbf{v}\ linking \overrightarrow{P}\ to \overrightarrow{Q}\ can be expressed as: \(a \mathbf{i} + b \mathbf{j}\). Where \mathbf{i}\ and \mathbf{j}\ make it clear we're discussing it in two dimensions.

coordinate geometry

Coordinate geometry, also known as analytic geometry, utilizes algebraic equations to represent geometric principles. It allows us to calculate distances and midpoints, find slopes, and determine vector components from given points.
Given points \(P = (-4, 3)\) and \(Q = (5, -1)\), we use coordinate geometry to find the vector form of \overrightarrow{PQ}\. To find this vector, we subtract the coordinates of \overrightarrow{P}\ from \overrightarrow{Q}\ resulting in the components \(a\) and \(b\).

• Component a: 5 - (-4) = 9
• Component b: -1 - 3 = -4

Thus, we now have the coordinates of the vector expressed in \mathbf{v} = 9 \mathbf{i} - 4 \mathbf{j}\.

magnitude of a vector

The magnitude of a vector, often referred to as its length or norm, tells us about the size of the vector. It's a measure of how long the vector is, irrespective of its direction.
For a vector given by \(a \mathbf{i} + b \mathbf{j}\), the magnitude \|\backslashmathbf{v}\backslash|\ is calculated using the Euclidean distance formula:
\|\mathbf{v}\| = \sqrt{a^2 + b^2}\
This formula comes from the Pythagorean theorem and applies to vectors in two dimensions. For our vector \( \mathbf{v} = 9 \mathbf{i} - 4 \mathbf{j} \):

• \a = 9\
• \b = -4\

Plugging in the values, we get:
\|\mathbf{v}\| = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97}\. This result gives us the magnitude of the vector, \sqrt{97}\, explaining how lengthy our vector \mathbf{v}\ is.

component form of a vector

To understand the vector in the context of movement or direction, representing it in its component form is essential. The component form breaks down the vector into its parts along the x and y axes. This helps in various vector operations such as addition, subtraction, and determining magnitude.
From the exercise, for points \(P = (-4, 3)\) and \(Q = (5, -1)\), the vector \mathbf{v}\ can be created by calculating the difference in their respective coordinates.

• Component along x (a): \5 - (-4) = 9\
• Component along y (b): \-1 - 3 = -4\

This leads us to the vector components \(a \mathbf{i} + b \mathbf{j} = 9 \mathbf{i} - 4 \mathbf{j}\). These components signify how much the vector moves along the x-axis and y-axis respectively.
Breaking down vectors in this way allows us to use them effectively in various mathematical and physical applications, from calculating forces to finding directions in navigation.