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Chapter 5: Problem 33

Find the exact magnitude and direction angle to the nearest tenth of a degree of each vector. $$ \langle 5,0\rangle $$

Short Answer

Expert verified
Magnitude: 5, Direction angle: 0 degrees.

Step by step solution

01

Identify the Components of the Vector

The given vector is \(\textbf{v} = \langle 5, 0 \rangle\). The components of the vector are 5 (in the x-direction) and 0 (in the y-direction).
02

Calculate the Magnitude of the Vector

The magnitude of a vector \(\textbf{v} = \langle a, b \rangle\) is given by the formula \(\textbf{|v|} = \sqrt{a^2 + b^2}\). Substituting the given components: \(\textbf{|v|} = \sqrt{5^2 + 0^2} = \sqrt{25} = 5\).
03

Determine the Direction Angle

The direction angle \(\theta\) of a vector is given by the formula \(\theta = \tan^{-1}(\frac{b}{a})\). For the vector \(\textbf{v} = \langle 5, 0 \rangle\), \(\theta = \tan^{-1}(\frac{0}{5})\). Since \(\tan^{-1}(0) = 0\), the direction angle is 0 degrees.
04

Conclusion

The exact magnitude of the vector is 5, and the direction angle is 0 degrees.

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Key Concepts

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

Vector Components
When working with vectors, it's important to understand their components. A vector in the plane is typically written as \(\) where \(a\) is the horizontal component (along the x-axis) and \(b\) is the vertical component (along the y-axis). For example, in the vector \(<5, 0>\), 5 is the x-component, and 0 is the y-component. These components tell us how far the vector moves in each direction. This helps break down the vector's overall movement into simpler, understandable parts.
Magnitude Formula
The magnitude of a vector, often thought of as its length, can be found using a simple formula. For a vector \(\), the magnitude \(|\textbf{v}|\) is given by \sqrt{a^2 + b^2}\. This formula comes from the Pythagorean theorem, which works perfectly because a vector forms a right triangle with its components.

Let's check the vector \(<5, 0>\) again. To find its magnitude, plug the components into the formula: \sqrt{5^2 + 0^2} = \sqrt{25} = 5\. So, the magnitude of this vector is 5. This tells us how long the vector is from the origin to its end point.
Direction Angle Formula
The direction angle of a vector is the angle that it makes with the positive x-axis. This angle is very important because it tells us the vector's orientation in the plane.

To find this angle, we use the formula \theta = \tan^{-1}(\frac{b}{a})\. It's based on the arctangent function, which helps determine the angle from a ratio of the sides of a triangle.

In the case of our vector \(<5, 0>\), we calculate \theta = \tan^{-1}(\frac{0}{5}) = \tan^{-1}(0)\. Since \tan^{-1}(0) = 0\, the direction angle is 0 degrees. This means the vector points directly along the positive x-axis.

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Most popular questions from this chapter

Determine whether each pair of vectors is parallel, perpendicular, or neither. $$ \langle 2,-4\rangle,\langle 2,1\rangle $$

Prove that scalar multiplication is distributive over vector addition, first using the component form and then using a geometric argument.

Given that \(\mathbf{A}=\langle 3,1\rangle\) and \(\mathbf{B}=\langle-2,3\rangle,\) find the magnitude and direction angle for each of the following vectors. Give exact answers using radicals when possible. Otherwise round to the nearest tenth. $$ -\mathbf{A}+\frac{1}{2} \mathbf{B} $$

Find the component form for each vector \(\mathbf{v}\) with the given magnitude and direction angle \(\theta .\) Give exact values using radicals when possible. Otherwise round to the nearest tenth. $$ |\mathbf{v}|=290, \theta=145^{\circ} $$

Write each vector as a linear combination of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$ \langle 1,1\rangle $$
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