Lines and Line Segments Find parametric equations for the lines in Exercises 1 -12. 1. The line through the point \( P(3,-4-1) \) parallel to the vector \( i+j+k \)

To find the parametric equations for the line through the point \( P(3, -4, -1) \) that is parallel to the vector \( \mathbf{i} + \mathbf{j} + \mathbf{k} \), we can utilize the point-direction form of the line. The vector \( \mathbf{v} = \langle 1, 1, 1 \rangle \) indicates that as \( t \) (the parameter) changes, all three coordinates change equally. Therefore, the parametric equations for the line can be expressed as: \[ x = 3 + t, \quad y = -4 + t, \quad z = -1 + t \] As \( t \) varies over all real numbers, these equations describe the line extending in both directions through point \( P \). In the world of geometry, parametric equations like these are super useful! Think of them as a roadmap for points that lie on your line. They allow you to not just find coordinates, but also visualize motion along that line. So next time you’re plotting or analyzing geometric shapes or lines, remember that these equations let you traverse along a path as simply as adjusting the radio dial in your car!