Problem 14 Find the magnitude and direction... [FREE SOLUTION]

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Trigonometry

Cynthia Y. Young

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 7: Problem 14

Find the magnitude and direction angle of each vector. $$ \mathbf{u}=\langle 0,7\rangle $$

Short Answer

Expert verified

Magnitude is 7, direction angle is $90^\circ$.

Step by step solution

Understand the Problem

We are given the vector $\mathbf{u} = \langle 0, 7 \rangle$ and we need to find its magnitude and the direction angle. The vector has components: $x = 0$ and $y = 7$.

Calculate the Magnitude

The magnitude of a vector $\mathbf{u} = \langle x, y \rangle$ is calculated using the formula $\| \mathbf{u} \| = \sqrt{x^2 + y^2}$. For $\mathbf{u} = \langle 0, 7 \rangle$, this becomes: \[ \| \mathbf{u} \| = \sqrt{0^2 + 7^2} = \sqrt{49} = 7\] So, the magnitude of $\mathbf{u}$ is 7.

Calculate the Direction Angle

The direction angle ($\theta$) of a vector is found using the tangent formula $\tan(\theta) = \frac{y}{x}$. Since $x = 0$, the vector is purely vertical. The angle for a vector pointing straight up is $90^\circ$ (or $\frac{\pi}{2}$ radians). If the vector were pointing down, the angle would be $270^\circ$ (or $\frac{3\pi}{2}$ radians). In this case, since $y = 7$ and positive, it is indeed pointing straight up, so $\theta = 90^\circ$.

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Key Concepts

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

Magnitude of a Vector

The magnitude of a vector can be thought of as the length of the vector. It's a way to describe how "intense" or "strong" the vector is. For any vector $ \mathbf{u} = \langle x, y \rangle $, the magnitude $ \| \mathbf{u} \| $ is calculated by the formula:

$ \| \mathbf{u} \| = \sqrt{x^2 + y^2} $

For our specific vector $ \mathbf{u} = \langle 0, 7 \rangle $, it means

$ \| \mathbf{u} \| = \sqrt{0^2 + 7^2} = \sqrt{49} = 7 $

This tells us that the vector is 7 units long.
It's just like measuring the length of a stick or line on a coordinate plane, even though it's in a different direction.

Direction Angle

The direction angle of a vector is the angle the vector makes with the positive x-axis. It helps us understand which way the vector is pointing. To find the direction angle, $ \theta $, of a vector $ \mathbf{u} = \langle x, y \rangle $, we typically use the tangent formula:

$ \tan(\theta) = \frac{y}{x} $

Because our vector $ \mathbf{u} = \langle 0, 7 \rangle $ has $ x = 0 $, it's a special case. The vector doesn't "go" anywhere along the x-axis.
It's purely vertical.
For vertical vectors:

If $ y > 0 $, $ \theta = 90^\circ $ (or $ \frac{\pi}{2} $ radians), pointing straight up.
If $ y < 0 $, $ \theta = 270^\circ $ (or $ \frac{3\pi}{2} $ radians), pointing straight down.

Since in our case $ y = 7 $ which is positive, the direction angle $ \theta $ is $ 90^\circ $, indicating the vector points straight upwards.

Vector Components

Vector components are like the building blocks of a vector. Any vector can be broken down into its components along the x-axis and the y-axis.
For the vector $ \mathbf{u} = \langle x, y \rangle $, the components are simply $ x $ and $ y $.

The x-component shows how far left or right the vector moves.
The y-component shows how far up or down the vector moves.

In our example with the vector $ \mathbf{u} = \langle 0, 7 \rangle $:

The x-component is 0, meaning no movement left or right.
The y-component is 7, meaning it moves completely upwards by 7 units.

Understanding components helps deconstruct where and how a vector moves. It's vital in physics and engineering for determining forces, velocities, and more.

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91影视

Find the magnitude and direction angle of each vector. $$ \mathbf{u}=\langle 0,7\rangle $$

Short Answer

Step by step solution

Understand the Problem

Calculate the Magnitude

Calculate the Direction Angle

Key Concepts

Magnitude of a Vector

Direction Angle

Vector Components

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