The shortest distance between the lines [ r=(13 i+2 j+83 k)+(2 i-5 j+6 k) and r=(-23 i-13 …

Question

The shortest distance between the lines
$$\vec r=\left(\frac13\hat i+2\hat j+\frac83\hat k\right)+\lambda(2\hat i-5\hat j+6\hat k) \quad\text{and}\quad \vec r=\left(-\frac23\hat i-\frac13\hat k\right)+\mu(\hat j-\hat k),\quad \lambda,\mu\in\mathbb R,$$
is:

Accepted Answer

$\vec a_1=\left(\tfrac13,2,\tfrac83\right),\ \vec b_1=(2,-5,6)$; $\vec a_2=\left(-\tfrac23,0,-\tfrac13\right),\ \vec b_2=(0,1,-1)$.

$$\vec a_2-\vec a_1=(-1,-2,-3).$$
$$\vec b_1\times\vec b_2=\hat i[(-5)(-1)-6(1)]-\hat j[2(-1)-6(0)]+\hat k[2(1)-(-5)(0)]=(-1,2,2).$$
$$|\vec b_1\times\vec b_2|=\sqrt{1+4+4}=3.$$
$$(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)=(-1)(-1)+(-2)(2)+(-3)(2)=1-4-6=-9.$$
$$d=\frac{|-9|}{3}=3.$$

Solutions · 1