It's a question for vectors about angles

Question

Line L makes a 120 degree with the positive x-axis, a 45-degree angle with the positive y-axis and a 60 degree angle with the positive z-axis. Write a symmetric equation for Line L.

I'm not sure what to do with the angles.

Answer 1

Hello. We need to use the equation cos(theta) = v • w / magnitude of v * magnitude of w.

To continue, we need to realize that the x-axis vector is (x,0,0) where x is some number, y-axis vector is (0,y,0) where y is some number, x-axis vector is (0,0,z) where z is some number. We also can create direction for Line L: (x1,y1,z1) these are non-related to the previous variables. Now when we substitute everything into the equation we get: Cos(120) = x * x1 / x * root (x1^2 * y1^2 * z1^2), Cos(45) = y * y1 / y * root (x1^2 * y1^2 * z1^2), Cos(60) = z * z1 / z * root (x1^2 * y1^2 * z1^2). After simplifying we get, Cos(120) = x1 /w, Cos(45) = y1 /w, Cos(60) = z1 /w, were w is root (x1^2 * y1^2 * z1^2).

From here we get the equation of the directions by setting everything equal to each other:

x1/ cos (120 ) = y1/ cos (45 ) = z1/ cos (60). This is the symmetrical equation of Line L.The Vector Equation of a Line can be shown too. Given Line L intersects with all axis we know it passes through the origin of (0,0,0). Therefore Line L = (0, 0, 0) + S ( -2, 1.41, 2 ), were S is the parameter. Note: I wanted to give my best answer, and am not 100% confident.