If the vector a = a(1)i + a(2)j + a(3)k makes angles alpha, beta and gamma respectively with the positive directions (measured anti clockwise) of the x, y and z axes then how do you derive the formulas:
cos alpha= a(1)/|a|, cos beta= a(2)/|a| and cos gamma= a(3)/|a|??
cos alpha= a(1)/|a|, cos beta= a(2)/|a| and cos gamma= a(3)/|a|??
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a = a(1)i + a(2)j + a(3)k
As you know
cos alpha, cos beta and cos gamma are direction cosines.
Also, from dot product of two vectors
a.b=|a||b|cos(theta)
Thus,
a.x = (a(1)i + a(2)j + a(3)k ). i = a
so, cos alpha = a/|a|
Similarly,
cos beta = a2/|a| and cos gamma = a3/|a|
As you know
cos alpha, cos beta and cos gamma are direction cosines.
Also, from dot product of two vectors
a.b=|a||b|cos(theta)
Thus,
a.x = (a(1)i + a(2)j + a(3)k ). i = a
so, cos alpha = a/|a|
Similarly,
cos beta = a2/|a| and cos gamma = a3/|a|