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Illustration Vectors and vector components - orthogonal coordinate system

Vectors and vector components - orthogonal coordinate system
Vectors and vector components - orthogonal coordinate system
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You see an orthogonal coordinate system with coordinates \((x,~y,~z)\) any two three-dimensional vectors \(\boldsymbol{a} \) and \(\boldsymbol{b} \) and their components:\[ \boldsymbol{a} ~=~ (a_1,~a_2,~a_3) \] \[ \boldsymbol{b} ~=~ (b_1,~b_2,~b_3) \]

Here \(a_1\) is the length of the vector \( \boldsymbol{a} \) in the \(x\)-direction, \(a_2\) is the length of the vector \( \boldsymbol{a} \) in the \(y\)-direction, and \(a_3\) is the length of the vector \( \boldsymbol{a} \) in the \(z\)-direction. For the vector \( \boldsymbol{b} \) this applies analogously.