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Chapter 9: Problem 36

Find the magnitude and direction of \(\overrightarrow{C D}\) for the given coordinates. \(C(-2,1), D(2,5)\)

Short Answer

Expert verified
Magnitude: \(4\sqrt{2}\); Direction: \(45^\circ\).

Step by step solution

01

Identify the Components

First, identify the horizontal and vertical components of the vector \(\overrightarrow{CD}\). The change in the \(x\)-coordinates is \(x_2 - x_1 = 2 - (-2) = 4\). The change in the \(y\)-coordinates is \(y_2 - y_1 = 5 - 1 = 4\). Thus, \(\overrightarrow{CD}\) is described by the vector \([4, 4]\).
02

Calculate the Magnitude

The magnitude of a vector \(\overrightarrow{AB}\) with components \((x, y)\) is given by \(\sqrt{x^2 + y^2}\). For \(\overrightarrow{CD} = [4, 4]\), compute \(\sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\). Hence, the magnitude of \(\overrightarrow{CD}\) is \(4\sqrt{2}\).
03

Find the Direction

The direction of a vector is the angle \(\theta\) it makes with the positive \(x\)-axis, given by \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). For \(\overrightarrow{CD} = [4, 4]\), calculate \(\theta = \tan^{-1}\left(\frac{4}{4}\right) = \tan^{-1}(1)\). This gives \(\theta = 45^\circ\). The direction is \(45^\circ\) from the positive \(x\)-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
Understanding vector components is crucial when dealing with vectors in coordinate geometry. A vector can be broken down into two parts: the horizontal component and the vertical component.
  • The horizontal component is the change along the x-axis.
  • The vertical component is the change along the y-axis.
In the exercise, the vector \(\overrightarrow{CD}\) is defined by its endpoints: \(C(-2,1)\) and \(D(2,5)\). To find the components, calculate the difference between the coordinates:
  • Horizontal component: \(x_2 - x_1 = 2 - (-2) = 4\).
  • Vertical component: \(y_2 - y_1 = 5 - 1 = 4\).
Therefore, the vector \(\overrightarrow{CD}\) can be represented by \([4, 4]\). Breaking down vectors into these components helps us analyze and manipulate them mathematically.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of geometry that uses the coordinate system. It combines geometry and algebra to represent and solve equations graphically.
  • Points are described by coordinates, like \((x, y)\) in a 2D plane.
  • Vectors, like \(\overrightarrow{CD}\), have both magnitude and direction and can be described using their components in this system.
In coordinate geometry, vectors are often used to connect points with direction and magnitude. For example, the vector \(\overrightarrow{CD} = [4, 4]\) is not just any line but a directed line segment from point C to D. This method allows us to use algebraic equations to compute geometric shapes and solve problems, like finding distances and angles.
Magnitude of a Vector
The magnitude of a vector tells us about its length or size and is a crucial aspect when working with vectors.Let's consider the vector \([4, 4]\) from our exercise. The formula to find the magnitude \(\|\overrightarrow{v}\|\) of a vector \(\overrightarrow{v} = [x, y]\) is \(\sqrt{x^2 + y^2}\). Applying this to our vector:
  • Calculate each component squared: \(4^2 = 16\),
  • Add them together: \(16 + 16 = 32\),
  • Take the square root: \(\sqrt{32} = 4\sqrt{2}\).
Therefore, the magnitude of \(\overrightarrow{CD}\) is \(4\sqrt{2}\). The magnitude gives the "length" of the vector and is always a non-negative number. It's an essential part of understanding the scale of the vector.
Direction of a Vector
The direction of a vector is vital to understand where the vector is pointing. It's usually expressed as an angle with respect to a reference direction, typically the positive x-axis in the Cartesian coordinate system.To find the direction, use the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Here's how it works for our vector:
  • Substitute the components: \(\theta = \tan^{-1}\left(\frac{4}{4}\right)\).
  • This simplifies to \(\tan^{-1}(1)\), which equals 45掳.
Thus, the vector \(\overrightarrow{CD}\) makes a 45掳 angle with the positive x-axis.Understanding both the magnitude and direction of a vector allows us to have a complete description of its orientation and length. This helps in various applications, such as physics, engineering, and computer graphics.

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