In a cube with edge length , find the cosine of the angle between the skew lines and .

The cosine of the angle between and is .

Solid Geometry Advanced Solution – Problem 8: find the cosine of the angle between the skew lines and

Question

In a cube $ABCD-A_1B_1C_1D_1$ with edge length $2$ , find the cosine of the angle between the skew lines $AC_1$ and $BD_1$ .

Step-by-step solution

(1) Use coordinates: $A(0,0,0),\ B(2,0,0),\ C(2,2,0),\ D(0,2,0),\ C_1(2,2,2),\ D_1(0,2,2).$ (2) Direction vectors: $\mathbf u=\overrightarrow{AC_1}=(2,2,2),\qquad \mathbf v=\overrightarrow{BD_1}=D_1-B=(-2,2,2).$ (3) Then $\cos\theta=\frac{|\mathbf u\cdot\mathbf v|}{|\mathbf u||\mathbf v|}.$ Compute $\mathbf u\cdot\mathbf v=(2)(-2)+(2)(2)+(2)(2)=4,$ and $|\mathbf u|=|\mathbf v|=2\sqrt3$ . Thus $\cos\theta=\frac{4}{(2\sqrt3)(2\sqrt3)}=\frac{1}{3}.$

Final answer

The cosine of the angle between $AC_1$ and $BD_1$ is $\dfrac{1}{3}$ .

Marking scheme

Step 1 — Setup

Checkpoint: set coordinates for the cube and write direction vectors $\mathbf u,\mathbf v$ (2 pts)

Step 2 — Key Calculation

Checkpoint: compute $\mathbf u\cdot\mathbf v$ and norms correctly (3 pts)

Step 3 — Final Answer

Checkpoint: simplify to $\cos\theta=\frac13$ (2 pts)

Zero credit if: uses endpoints to compute an angle between segments that are not direction vectors.

Deductions: -1 pt for arithmetic error in dot product.

Solid Geometry Advanced – Problem 8: find the cosine of the angle between the skew lines and