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 ?
You are right, I wasn't watching the order, and it was the angle with the x-axis that we didn't have.
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?
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)?
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?
To determine the number of lines through the origin that make angles of 60 degrees with both the +y and +z axes, we can visualize a 3-dimensional coordinate system.
First, let's consider the +y axis. Any line passing through the origin and making an angle of 60 degrees with the +y axis will lie in a plane perpendicular to the +y axis. This plane can be represented by a line passing through the origin in the +x and +z directions.
Similarly, for the +z axis, any line passing through the origin and making an angle of 60 degrees with the +z axis will lie in a plane perpendicular to the +z axis. This plane can be represented by a line passing through the origin in the +x and +y directions.
To find the lines that satisfy both conditions, we need to find the intersection of these two planes. Since both planes share the same line passing through the origin in the +x direction, the intersection line will also lie along this common line.
Therefore, there is only one line that passes through the origin and makes angles of 60 degrees with both the +y and +z axes.
Now, let's find the angle this line makes with the +x axis. Since the line lies in the plane formed by the +x and +y axes, its projection onto the +xy plane will be perpendicular to the +x axis. Hence, the angle it makes with the +x axis is 90 degrees.
In summary:
- There is only one line passing through the origin that makes angles of 60 degrees with both the +y and +z axes.
- This line makes an angle of 90 degrees with the +x axis.