Let be the line through in direction , and let be the line through in direction . Find the distance between the skew lines and .

Solid Geometry Advanced Solution – Problem 12: Find the distance between the skew lines and

Question

Let $l_1$ be the line through $A(0,0,0)$ in direction $\mathbf u=(1,1,0)$ , and let $l_2$ be the line through $B(0,1,1)$ in direction $\mathbf v=(1,0,1)$ .

Find the distance between the skew lines $l_1$ and $l_2$ .

Step-by-step solution

(1) For skew lines, the distance is $d=\frac{|\overrightarrow{AB}\cdot(\mathbf u\times\mathbf v)|}{|\mathbf u\times\mathbf v|},\quad \overrightarrow{AB}=B-A=(0,1,1).$ (2) Compute the cross product: $\mathbf u\times\mathbf v=(1,1,0)\times(1,0,1)=(1,-1,-1).$ So $|\mathbf u\times\mathbf v|=\sqrt3$ .

(3) Scalar triple product: $\overrightarrow{AB}\cdot(\mathbf u\times\mathbf v)=(0,1,1)\cdot(1,-1,-1)=-2.$ Thus $d=\frac{|-2|}{\sqrt3}=\frac{2}{\sqrt3}.$

Final answer

The distance between $l_1$ and $l_2$ is $\dfrac{2}{\sqrt3}$ .

Marking scheme

Step 1 — Setup

Checkpoint: state the skew-line distance formula using the scalar triple product (2 pts)

Step 2 — Key Calculation

Checkpoint: compute $\mathbf u\times\mathbf v$ and $\overrightarrow{AB}\cdot(\mathbf u\times\mathbf v)$ correctly (3 pts)

Step 3 — Final Answer

Checkpoint: conclude $d=\frac{2}{\sqrt3}$ (2 pts)

Zero credit if: treats the lines as intersecting and sets the distance to 0.

Deductions: -1 pt for forgetting absolute value in the triple product.

Solid Geometry Advanced – Problem 12: Find the distance between the skew lines and