Finding Perpendiculars

• Take away one parallel vector from another
• Write a new vector in terms of \(\lambda-\mu\) (\(t\)) (This vector is perpendicular to the other two vectors)
• Multiply this vector out to get a single vector. (? + ?t)
• If we multiply this vector with the direction vector of our parallel lines, we get 0.
• With this info, we can solve for \(t\)
• Once we solve for \(t\), we can sub this back in to our original perpendicular equation and find an equation for the vector of the distance between the two lines.
• You can then use the magnitude equation \(\sqrt{ a^2+b^2 +c^2}=\text{magnitude}\)
• We now have the distance bwteen the two vectors.

• Take away one parallel vector from another.
• Write a new vector in the form \((?+?\lambda+?\mu)\)
• Call this vector \(AB\)
• \(AB\times \text{direction vector 1}=0\)
• \(AB\times \text{direction vector 2}=0\)
• We can multiply this out and get two simultaneous equations to solve in terms of \(\mu\) and \(\lambda\).
• By eliminating either of these variables we can find the other and the sub this back in to find the other.
• Finally we go back to our original perpendicular equation in terms of \(\mu\) and \(\lambda\).
• This gives us a final vector which we use the magnitude equation \(\sqrt{ a^2+b^2 +c^2}=\text{magnitude}\)
• We now have the distance between the two vectors.

• Multiply out your vector into a single vector equation.
• Find the perpendicular equation by inverting the signs on the point vector and adding this to our single vector equation we multiplied out.
• Call it AB
• \(AB\times \text{direction vector}=0\)
• Find \(\lambda\)
• Find the full form of AB.
• Find the magnitude.

\(\(\mathbf{r}.\mathbf{\hat{n}}=\mathbf{d}\)\)
- Where \(\mathbf{r}\) is any point on our plane.
- Where \(\mathbf{\hat{n}}\) is the normal vector of the plane.
- Where \(\mathbf{d}\) is the shortest distance between a plane and the origin.

\(\(\frac{\mathbf{x}.\mathbf{n}-\mathbf{d}}{|\mathbf{n}|}\)\)
- Where \(\mathbf{x}\) is the vector of the point.
- Where \(\mathbf{n}\) is the normal vector of the plane.
- Where \(\mathbf{d}\) is the constant of the plane.
- Where \(|\mathbf{n}|\) is the magnitude of the normal vector.