MHB Find Direction Angles: Calculate & Understand

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SUMMARY

The discussion focuses on calculating direction angles of a vector in three-dimensional space. The vector's length is determined to be \(2\sqrt{2}\), derived from the components along the x, y, and z axes, specifically \(\sqrt{2}\), 2, and \(\sqrt{2}\). The cosine of the angle with the x-axis is calculated as \(\frac{1}{2}\), leading to an angle of 60 degrees. Participants are encouraged to apply the same method to find the angles with the y and z axes.

PREREQUISITES
  • Understanding of vector components in three-dimensional space
  • Knowledge of trigonometric functions, specifically cosine
  • Familiarity with the Pythagorean theorem
  • Ability to perform basic algebraic manipulations
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  • Calculate direction angles for vectors using different component values
  • Explore the relationship between direction angles and vector projections
  • Learn about the geometric interpretation of vectors in 3D space
  • Study the applications of direction angles in physics and engineering
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Students and professionals in mathematics, physics, and engineering who need to understand vector analysis and its applications in three-dimensional space.

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Do you know what "direction angles" are? They are the angles the vector makes with each of the coordinate axes. The length of this vector is $\sqrt{2+ 4+ 2}=\sqrt{8}= 2\sqrt{2}$. That is the length of the hypotenuse of the right triangles the vector makes with each of the axes. The length of the side along the x-axis has length $\sqrt{2}$. The cosine of that angle is $\frac{\sqrt{2}}{2\sqrt{2}}= \frac{1}{2}$. So what is that angle?

Do the other angles the same way.
 

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