/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 \(\|\mathbf{v}\|=4 \sqrt{3} \qua... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 62

\(\|\mathbf{v}\|=4 \sqrt{3} \quad \theta=90^{\circ}\)

Short Answer

Expert verified
The vector \(\mathbf{v}\) in Cartesian coordinates is \((0, 4 \sqrt{3})\).

Step by step solution

01

Understand Polar Coordinates

Vectors in an Euclidean plane can be expressed either in Cartesian coordinates \((x, y) = (r \cos(\theta), r \sin(\theta))\), where \(r\) is the modulus (magnitude) and \(\theta\) is the direction angle from the positive x-axis, or in Polar coordinates \((r, \theta)\).
02

Apply Conversion Formulas

To convert from Polar to Cartesian coordinates, we have to use the conversion formulas:\n\n\(x = r \cos(\theta) = \|\mathbf{v}\| \cos(\theta)\)\n\nand\n\n\(y = r \sin(\theta) = \|\mathbf{v}\| \sin(\theta)\)\n
03

Substitute Given Values

Substitute the given values into the conversion formulas: \(\|\mathbf{v}\| = 4 \sqrt{3}\) and \(\theta = 90^{\circ} = \pi/2 \) (in radians). This gives us \(x = 4 \sqrt{3} \cos(\pi/2) = 0\) and \(y = 4 \sqrt{3} \sin(\pi/2) = 4 \sqrt{3}\).
04

Write Final Answer

The vector \(\mathbf{v}\) in Cartesian coordinates is \((0, 4 \sqrt{3})\).

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Key Concepts

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

Polar Coordinates
Polar coordinates is a system that lets us represent points in the plane using a distance and an angle. Imagine standing at the origin of a graph. You start facing along the positive x-axis.
  • **Distance from Origin:** This is known as the magnitude or radius, noted as \( r \). It tells you how far away you are from the origin.
  • **Direction/Angle:** Denoted as \( \theta \), it indicates the direction you need to turn from the positive x-axis to reach the point.
In our exercise, we are given a vector of magnitude \( \|\mathbf{v}\| = 4 \sqrt{3} \) and an angle \( \theta = 90^{\circ} \). The magnitude tells us the vector's length, and the angle determines its orientation.
Cartesian Coordinates
The Cartesian coordinate system uses pairs (or triples for 3D) of coordinates to specify points in the plane (or space). These coordinates are based on distances measured along axes at 90 degrees to each other.
  • The **x-coordinate** tells you how far to move horizontally from the origin.
  • The **y-coordinate** tells you how much to move vertically.
In our exercise, we started with polar coordinates and converted to Cartesian coordinates using specific formulas. These formulas are:
  • \( x = r \cos(\theta) \)
  • \( y = r \sin(\theta) \)
Given \( \|\mathbf{v}\| = 4 \sqrt{3} \) and \( \theta = 90^{\circ} \), these convert to \( x = 0 \) and \( y = 4 \sqrt{3} \), showing that our vector lies entirely on the y-axis.
Vector Magnitude
A vector's magnitude is the length of the vector. It measures how far the vector stretches in the plane or space.
  • Magnitude is always a non-negative value.
  • The formula to find the magnitude of a vector in Cartesian coordinates \((x, y)\) is \( \sqrt{x^2 + y^2} \).
However, in polar coordinates, the magnitude is given directly as \( r \). In our context, the vector's magnitude is \( 4 \sqrt{3} \), which tells us the vector's length regardless of its direction. Magnitude is crucial since it affects the resulting values in conversions between coordinate systems.

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