Question

In Exercises 7-10, points \(P\) and \(Q\) are given. Write the vector \(\overrightarrow{P Q}\) in component form and using the standard unit vectors. \(P=(2,-1), \quad Q=(3,5)\)

Short answer

The vector \(\overrightarrow{PQ}\) is \((1, 6)\) or \(1\mathbf{i} + 6\mathbf{j}\).

Step-by-step solution

Understand the Problem

We are tasked with finding the vector \(\overrightarrow{PQ}\) given two points \(P\) and \(Q\) in a two-dimensional plane. Point \(P\) is \((2,-1)\) and point \(Q\) is \((3,5)\). This vector can be expressed in component form as well as using standard unit vectors.

Calculate the Components of the Vector

The vector \(\overrightarrow{PQ}\) has its components calculated as the difference between the corresponding coordinates of \(Q\) and \(P\). The formula for the component form of the vector \(\overrightarrow{PQ}\) is \((x_2-x_1, y_2-y_1)\), where \(x_1\) and \(y_1\) are coordinates of \(P\) and \(x_2\) and \(y_2\) are coordinates of \(Q\). Here, the vector \(\overrightarrow{PQ} = (3-2, 5-(-1)) = (1, 6)\).

Express the Vector using Standard Unit Vectors

The component form \((1, 6)\) can be expressed using the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). The vector \(\overrightarrow{PQ} = 1\mathbf{i} + 6\mathbf{j}\) where \(\mathbf{i}\) is the unit vector in the x-direction and \(\mathbf{j}\) is the unit vector in the y-direction.

Key concepts

Component Form

In vector calculus, expressing a vector in component form means identifying its parts based on direction and magnitude in a coordinate system. Consider a two-dimensional vector represented by the points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \). The vector \( \overrightarrow{PQ} \) is described by its horizontal and vertical components, calculated as the differences in the coordinates:

• Horizontal component: \( x_2 - x_1 \)
• Vertical component: \( y_2 - y_1 \)

The equation for \( \overrightarrow{PQ} \), given the points \( P(2, -1) \) and \( Q(3, 5) \), becomes:

• Horizontal component: \( 3 - 2 = 1 \)
• Vertical component: \( 5 - (-1) = 6 \)

Thus, the vector \( \overrightarrow{PQ} \) is \( (1, 6) \) in component form. This offers a simple way to describe vectors using easily understood parts corresponding to the x and y axes.

Standard Unit Vectors

Once you have a vector in component form, you can further express it using standard unit vectors, simplifying vector operations. Standard unit vectors include:

• \( \mathbf{i} \), the unit vector along the x-axis
• \( \mathbf{j} \), the unit vector along the y-axis

For the vector \( \overrightarrow{PQ} = (1, 6) \), you can represent it with standard unit vectors by scaling \( \mathbf{i} \) and \( \mathbf{j} \) with the components:

• \( \overrightarrow{PQ} = 1\mathbf{i} + 6\mathbf{j} \)

This method makes it easier to visualize and compute operations with vectors. Essentially, the vector \( \overrightarrow{PQ} \) puts a spotlight on the contribution of each axis direction—x and y—in the vector's overall trajectory.

Two-Dimensional Vectors

Two-dimensional vectors are foundational in vector calculus, existing in a plane defined by two axes: x and y. These vectors illustrate direction and magnitude within a two-dimensional space, stemming from physics and engineering problems.In the given points \( P(2, -1) \) and \( Q(3, 5) \), the vector \( \overrightarrow{PQ} \) falls in this category. Such vectors are essential building blocks as they:

• Enable the breakdown of motion into simplified linear components
• Allow for the addition and subtraction of vectors with geometric meaning
• Simplify complex calculations in navigation and trajectory planning

Understanding two-dimensional vectors facilitates grasping more advanced concepts in multi-dimensional spaces, providing a sturdy groundwork for further vector applications.