Write a vector equation for the line through \((1,2)\) with direction \((3,-1)\).

Find the point on \(\mathbf r=(1,2)+\lambda(3,-1)\) when \(\lambda=2\).

Does \((7,0)\) lie on \(\mathbf r=(1,2)+\lambda(3,-1)\)?

Find a direction vector for the line through \(A=(1,4,0)\) and \(B=(3,1,5)\).

Write a line through \(A=(1,4,0)\) and \(B=(3,1,5)\).

Find where \(\mathbf r=(2,1)+t(1,3)\) meets the \(y\)-axis.

Find where \(\mathbf r=(2,1)+t(1,3)\) meets the line \(y=7\).

Are lines \(\mathbf r=(0,1)+s(2,4)\) and \(\mathbf r=(3,-1)+t(1,2)\) parallel?

Find the intersection of \((0,0)+s(1,2)\) and \((3,0)+t(-1,1)\).

Find the distance from \(C=(1,2,3)\) to the line \(\mathbf r=(0,0,0)+\lambda(2,0,1)\).

Find a line through \((1,1,1)\) parallel to \((2,-1,0)\).

Find \(t\) if \((5,-1,1)\) lies on \(\mathbf r=(1,1,1)+t(2,-1,0)\).

Find the closest point on \(\mathbf r=(0,0)+t(1,0)\) to \(C=(3,4)\).

Write symmetric equations for \(\mathbf r=(1,2,3)+t(2,-1,4)\).

Find the angle between lines with directions \((1,0,1)\) and \((0,1,1)\).

Do \(\mathbf r=(1,0,0)+s(1,1,0)\) and \(\mathbf r=(0,1,1)+t(1,1,0)\) represent the same line?

Find a parameter value where \(\mathbf r=(2,3,4)+t(1,-1,2)\) has \(z=10\).

Find the line through \((2,0,-1)\) parallel to \((1,-1,2)\), and show that this line is parallel to the plane with normal \((1,1,0)\).

Decide whether \((4,6)\) lies on the segment from \((1,0)\) to \((5,8)\), then find the ratio in which it divides the segment.

Find the shortest distance between the parallel lines \(\ell_1:(0,0)+s(1,0)\) and \(\ell_2:(0,3)+t(1,0)\), and identify one pair of closest points.