In a right triangular prism , the base is an equilateral triangle with side , and . Find the distance between the skew edges and .

The distance between skew edges and is .

Solid Geometry Solution – Problem 19: Find the distance between the skew edges and

Question

In a right triangular prism $ABC-A_1B_1C_1$ , the base $ABC$ is an equilateral triangle with side $2$ , and $AA_1=BB_1=CC_1=3$ . Find the distance between the skew edges $AB$ and $CC_1$ .

Step-by-step solution

Build coordinates with plane $ABC$ as $z=0$ : $A(0,0,0),\ B(2,0,0),\ C(1,\sqrt3,0),\ C_1(1,\sqrt3,3).$ Line $AB$ has direction $\vec d_1=\overrightarrow{AB}=(2,0,0)$ . Line $CC_1$ has direction $\vec d_2=\overrightarrow{CC_1}=(0,0,3)$ . Use the skew-line distance formula $d=\frac{|(\overrightarrow{AC})\cdot(\vec d_1\times\vec d_2)|}{|\vec d_1\times\vec d_2|}.$ Compute $\vec d_1\times\vec d_2=(2,0,0)\times(0,0,3)=(0,-6,0),$ and $\overrightarrow{AC}=(1,\sqrt3,0)$ . Then $|(\overrightarrow{AC})\cdot(\vec d_1\times\vec d_2)|=|(1,\sqrt3,0)\cdot(0,-6,0)|=6\sqrt3,$ so $d=\frac{6\sqrt3}{6}=\sqrt3.$

Final answer

The distance between skew edges $AB$ and $CC_1$ is $\sqrt3$ .

Marking scheme

1. Checkpoints (max 7 pts total)

Coordinate setup (2 pts): Place an equilateral base of side $2$ and height $3$ correctly.
Cross-product method (3 pts): Identify $\vec d_1,\vec d_2$ , compute $\vec d_1\times\vec d_2$ , and set up the skew-line distance formula.
Final computation (2 pts): Substitute and simplify to $\sqrt3$ .

2. Zero-credit items

Using a 2D distance in the base without justifying it as a skew-line distance.
Computing only $|\vec d_1\times\vec d_2|$ but not the triple product.

3. Deductions

Vector choice error (-1): using $\overrightarrow{CA}$ or a wrong connecting vector with incorrect sign handling.
Arithmetic error (-1): incorrect dot product value $6\sqrt3$ .

Solid Geometry – Problem 19: Find the distance between the skew edges and