Distance between two lines in a cube

[kate]

We have a cube ABCDA₁B₁C₁D₁ with edge a. Find the distance between the lines B₁C and BD.

[shyamjayakannan]

Let A be the origin. Then the coordinates of all the points are: A(0,0,0), B(a,0,0), C(a,a,0), D(0,a,0), A1(0,0,a), B1(a,0,a), C1(a,a,a), D1(0,a,a).
Now, B1C = [tex]\langle a,a,0\rangle+\langle0,-a,a\rangle t[/tex] and BD = [tex]\langle0,a,0\rangle+\langle a,-a,0\rangle t[/tex]. The distance between the lines will be the length of the normal vector between them. This can be found by taking the projection of the vector from C to D onto the common normal.

Normal vector = [tex]\langle0,-a,a\rangle\times\langle a,-a,0\rangle=\left\langle a^2,a^2,a^2\right\rangle[/tex] and DC = [tex]\langle a,0,0\rangle[/tex].

So, distance = [tex]\frac{\left\langle a^2,a^2,a^2\right\rangle\cdot\langle a,0,0\rangle}{\left|\left\langle a^2,a^2,a^2\right\rangle\right|}=\frac{a^2}{a^2\sqrt3}=\boxed{\frac{1}{\sqrt3}}[/tex]