1. **Problem Statement:** Determine the position of the given line relative to coordinate axes or other reference lines. 2. **Key Concepts:** The position of a line in 3D or 2D space can be described as: - Parallel to an axis or plane - Intersecting an axis or plane - Skew (not parallel and not intersecting) - Coincident (lying exactly on another line or axis) 3. **Formula and Approach:** - Use direction vectors and point coordinates of the line. - Check if the direction vector is parallel to any axis by comparing components. - Check if the line passes through the origin or intersects axes by substituting values. 4. **Step-by-step:** - Identify the direction vector of the line, say $\vec{d} = (d_x, d_y, d_z)$. - If two components of $\vec{d}$ are zero and one is nonzero, the line is parallel to the axis corresponding to the nonzero component. - If the line’s parametric equations satisfy the coordinate of an axis at some parameter value, it intersects that axis. - If the line does not intersect and is not parallel, it may be skew relative to other lines or planes. 5. **Example:** Suppose the line has parametric form: $$x = x_0 + t d_x, \quad y = y_0 + t d_y, \quad z = z_0 + t d_z$$ - To check if it is parallel to the x-axis, verify if $d_y = 0$ and $d_z = 0$. - To check intersection with y-axis, set $x=0$ and $z=0$ and solve for $t$; if $t$ exists, it intersects. 6. **Conclusion:** By analyzing the direction vector and parametric equations, you can determine the line’s position relative to axes or other lines. This method applies to all lines in the given diagrams for Homework 2.