Find the projection of the vector $\mathbf{v} = \langle 1,2,3\rangle$ onto the line $L: $ $x = 2+3t, y = 1+t, z = 4$

Determine if the following lines are parallel, skew, or intersecting. If they intersect, find the point of intersection. Otherwise, find the minimal distance between the two lines.

• $L_1: x = 1 + t, y = 2+3t, z = 1 - 2t$
• $L_2: x = 3 + 2t, y = -1+6t, z = -4t$
• $L_3: x = 4t, y = -1 -t, z = 3$

Construct three lines so that two are parallel but not the same and:

• The third line intersects both parallel lines.
• The third line intersects one, but is skew to the other.
• The third line is skew to both parallel lines.

Find three skew lines so that their projection onto the $xy$-plane forms an equliateral triangle.