Problem 31 - College Trigonometry

Question

Approximate the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Round your approximations to two decimal places. \(\|\vec{v}\|=26\); when drawn in standard position \(\vec{v}\) makes a \(304.5^{\circ}\) angle with the positive \(x\) -axis

Step-by-Step Solution

Answer

The component form of \(\vec{v}\) is approximately \((15.48, -19.90)\).

Step 1: Understand the Problem

We need to find the component form of the vector \(\vec{v}\) given its magnitude \(\|\vec{v}\| = 26\) and the direction angle \(\theta = 304.5^{\circ}\). The vector is in standard position, which means it originates from the origin.

Step 2: Find the Direction in Terms of Angle

The angle given is \(304.5^{\circ}\), which is measured counterclockwise from the positive \(x\)-axis. Since this is more than 270 degrees but less than 360 degrees, the vector lies in the fourth quadrant of the coordinate system.

Step 3: Use Trigonometric Functions to Find Components

The component form \(\vec{v}\) is given by \(\vec{v} = \langle \|\vec{v}\| \cos \theta, \|\vec{v}\| \sin \theta \rangle\). Using \(\|\vec{v}\| = 26\) and \(\theta = 304.5^{\circ}\), compute the components:- \(x\)-component: \(26 \cos(304.5^{\circ})\)- \(y\)-component: \(26 \sin(304.5^{\circ})\)

Key Concepts

Component Form

In vector mathematics, the component form of a vector refers to its breakdown into parts that align with a given coordinate system, usually denoted as the

• **x**-component
• **y**-component

. By expressing a vector in component form, we can easily manipulate and understand its behavior as it moves through a coordinate plane.
The process of finding the component form involves

• determining the vector's direction angle
• and its magnitude

. Once we have this information, we can apply trigonometric functions, like cosine and sine, to find how much of the vector's length is directed along each of the coordinate axes.
Specifically, the formula for the component form is given by \[\vec{v} = \langle \|\vec{v}\|\cos\theta,\|\vec{v}\|\sin\theta\rangle\] , where\(\|\vec{v}\|\) is the magnitude, and\(\theta\) is the direction angle of the vector. By translating a vector into its component form, we simplify many common vector operations like addition and subtraction.

Direction Angle

The direction angle of a vector is a critical component in understanding its orientation. This angle is typically measured from the positive**x**-axis in a counterclockwise direction.
It helps define which quadrant the vector lies in, which is helpful in vector calculus and physics when analyzing forces and motions.
In our example, the direction angle is\(304.5^{\circ}\). Since it is greater than\(270^{\circ}\) but less than\(360^{\circ}\), the vector lies in the fourth quadrant.
This positioning affects the signs of the vector's components:

• The x-component is positive
• The y-component is negative

.
Working with direction angles thoroughly helps in correctly applying trigonometric functions to determine each component's individual magnitude.

Magnitude

The magnitude of a vector, represented as\(\|\vec{v}\|\), describes the vector's length or size. Anyone dealing with vectors should understand how a vector's magnitude links to its ability to represent direction and force.
For instance, a larger magnitude implies a greater strength or effect in the given direction. In our problem, \(\|\vec{v}\|\)is equal to\(26\).
Understanding magnitude is crucial. This is especially true when it forms part of everyday physics problems, such as understanding forces, velocities, and accelerations. Without magnitude, vectors cannot be accurately represented or utilized. When breaking down vectors into component vectors, having the exact magnitude ensures correct calculations.
In mathematical terms, magnitude serves as the scalar that allows for the accurate breakdown of the direction and components of a vector as seen in the expression \[direction \theta \].
This assists in understanding the scale of the components themselves, such as converting the magnitude into horizontal and vertical components using trigonometric functions.

Trigonometric Functions

Trigonometric functions play an integral role in analyzing vectors. These functions, namely cosine and sine, convert a vector's magnitude and direction into sectional components along the Cartesian plane's axes.
This conversion is essential in visualizing and solving problems involving vector forces, especially when breaking down complex vector mechanics into simpler parts.

Using Cosine and Sine

The cosine and sine functions are particularly important:

• **Cosine** is used to find the horizontal component (adjacent) of the vector aligned to the**x**-axis. In formulaic terms, this is expressed as\(x = \|\vec{v}\| \cos\theta\)
• **Sine** measures the vertical component (opposite) lying along the**y**-axis. The representation would be\(y = \|\vec{v}\| \sin\theta\)

. Armed with the magnitude and angle, cosine and sine functions enable these calculations. Thus, they facilitate a clearer interpretation of vectors by breaking them down into manageable segments.