How many lines through the origin make angles of 60degree with both the +y and +z axes ? What angle do they make with the +x axis ?
5 answers
As usual, google can provide several discussions of the topic.
Let A, B, and C be the direction angles that our line makes with the x, the y, and the z axes respectively.
We know cos^2 A + cos^2 B + cos^2 C = 1
we are told that the line makes and angle of 60° with both the y and the z axes, and cos 60° = 1/2
so, cos^2 A + 1/4 + 1/4 = 1
cos^2 A = 1/2
cos A = 1/√2 or √2/2
then A = ± 45° , but -45° is coterminal with 315°
so the angle with the x-axis could be 45° or 315°
so our line could have terminal points of (1/2, 1/2, ±√2/2) or (1, 1, ±√2)
how many lines would that represent?
We know cos^2 A + cos^2 B + cos^2 C = 1
we are told that the line makes and angle of 60° with both the y and the z axes, and cos 60° = 1/2
so, cos^2 A + 1/4 + 1/4 = 1
cos^2 A = 1/2
cos A = 1/√2 or √2/2
then A = ± 45° , but -45° is coterminal with 315°
so the angle with the x-axis could be 45° or 315°
so our line could have terminal points of (1/2, 1/2, ±√2/2) or (1, 1, ±√2)
how many lines would that represent?
Reiny can you explain how you get (1, 1, ±√2). I don't understand how you get the terminal points in this order (1/2, 1/2, ±√2/2). If ( cos^2(A,cos^2(60),cos^(60)),should the terminal points be ( ±√2,1/2,1/2)?
You are right, I wasn't watching the order, and it was the angle with the x-axis that we didn't have.
Thank, So for (1/2, 1/2, ±√2/2) or (1, 1, ±√2), you just drop the denominator 2 because they all have a denominator of 2 and you get (±√2,1, 1 ). Is that right?
2 answers