N Neconine Nov 2009 11 0 May 24, 2010 #1 Not sure what to do here: The direction angles of a vector are all equal. Find the direction angles to the nearest degree.

Not sure what to do here: The direction angles of a vector are all equal. Find the direction angles to the nearest degree.

S Soroban MHF Hall of Honor May 2006 12,028 6,341 Lexington, MA (USA) May 24, 2010 #2 Hello, Neconine! The direction angles of a vector are all equal. Find the direction angles to the nearest degree. Click to expand... The direction angles of a vector are \(\displaystyle \alpha,\:\beta,\:\gamma\), which are the angles . . formed by the vector and the \(\displaystyle x.\:y,\) and \(\displaystyle z\) axes, respectively. Fact: .\(\displaystyle \cos^2\!\alpha + \cos^2\!\beta + \cos^2\!\gamma \:=\:1\) Since \(\displaystyle \alpha = \beta = \gamma\) we have: .\(\displaystyle \cos^2\!\alpha + \cos^2\!\alpha + \cos^2\!\alpha \;=\;1\) . . \(\displaystyle 3\cos^2\!\alpha \:=\:1 \quad\Rightarrow\quad \cos^2\!\alpha \:=\:\frac{1}{3} \quad\Rightarrow\quad \cos\alpha \:=\:\frac{1}{\sqrt{3}} \) . . \(\displaystyle \alpha \;=\;\cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \;=\;54.73561032^o \) Therefore: .\(\displaystyle \alpha \;=\;\beta\;=\;\gamma \;\approx\;55^o\)

Hello, Neconine! The direction angles of a vector are all equal. Find the direction angles to the nearest degree. Click to expand... The direction angles of a vector are \(\displaystyle \alpha,\:\beta,\:\gamma\), which are the angles . . formed by the vector and the \(\displaystyle x.\:y,\) and \(\displaystyle z\) axes, respectively. Fact: .\(\displaystyle \cos^2\!\alpha + \cos^2\!\beta + \cos^2\!\gamma \:=\:1\) Since \(\displaystyle \alpha = \beta = \gamma\) we have: .\(\displaystyle \cos^2\!\alpha + \cos^2\!\alpha + \cos^2\!\alpha \;=\;1\) . . \(\displaystyle 3\cos^2\!\alpha \:=\:1 \quad\Rightarrow\quad \cos^2\!\alpha \:=\:\frac{1}{3} \quad\Rightarrow\quad \cos\alpha \:=\:\frac{1}{\sqrt{3}} \) . . \(\displaystyle \alpha \;=\;\cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \;=\;54.73561032^o \) Therefore: .\(\displaystyle \alpha \;=\;\beta\;=\;\gamma \;\approx\;55^o\)