If a line makes angles of 90, 60, and 30 degrees with the positive direction

We assume that “positive direction” refers to the positive x-axis of a standard Cartesian coordinate system.

If a line makes angles of 90, 60, and 30 degrees with the positive direction, it means that the line makes angles of 90-90=0, 60-90=-30, and 30-90=-60 degrees with the positive x-axis.

To visualize this, we can draw a diagram as follows:

| /
| / 30 degrees
| /
| / * Line
| /
|/ 60 degrees
—*————–*——————
/ -60 degrees x-axis
/
/

Note that the line intersects the x-axis at some point. We can find the equation of the line using this point and its slope. To find the slope, we can use the fact that the tangent of an angle is equal to the opposite over the adjacent side of a right triangle.

For the 30-degree angle, the opposite side is the y-coordinate of the intersection point and the adjacent side is the x-coordinate. Thus, the slope of the line is:

tan(30) = y/x
y/x = sqrt(3)/3

For the 60-degree angle, the opposite side is the y-coordinate of the intersection point and the adjacent side is the distance from the intersection point to the y-axis, which is equal to the x-coordinate. Thus, the slope of the line is:

tan(60) = y/x
y/x = sqrt(3)

Since the line passes through the point of intersection (x, y), we can use the point-slope form of a line to write its equation:

y – y = m(x – x)
y – y = (sqrt(3)/3)(x – x)
y – y = (sqrt(3)/3)x

y – y = m(x – x)
y – y = sqrt(3)(x – x)
y – y = sqrt(3)x

Therefore, the equation of the line is either y = (sqrt(3)/3)x or y = sqrt(3)x.