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Question Number 60085 by mr W last updated on 17/May/19

Commented by mr W last updated on 17/May/19

$t h e e d g e l e n g t h o f c u b e i s a .$ $f i n d t h e m i n i m u m d i s t a n c e b e t w e e n$ $t h e b l u e l i n e s .$
Commented by ajfour last updated on 17/May/19

Commented by ajfour last updated on 17/May/19

$back left corner be origin .$ $x axis rightwards, y axis upwards .$ $r_{1} = λ a (i + j)$ $r_{2} = ai + μ (k - i)$ $d_{\min} = \frac{ai . (i + j) \times (k - i)}{∣ (i + j) \times (k - i) ∣}$ $= \frac{a}{\sqrt{3}} i . (- j + i + k) = \frac{a}{\sqrt{3}} .$
Commented by mr W last updated on 17/May/19

$c l e a r a n d s i m p l e w a y,$ $t h a n k y o u s i r!$ $l e t m e t r y w i t h o u t u s i n g v e c t o r s .$
	Answered by mr W last updated on 18/May/19

	Commented by mr W last updated on 18/May/19
	$A(a,a,0) B(a,0,a) eqn. of line AB: ((x−a)/(a−a))=((y−a)/(0−a))=((z−0)/(a−0))=λ or { ((x=a)),((y=a(1−λ))),((z=aλ)) :} C(a,a,a) D(0,0,a) eqn. of line CD: ((x−a)/(0−a))=((y−a)/(0−a))=((z−a)/(a−a))=μ or { ((x=a(1−μ))),((y=a(1−μ))),((z=a)) :} D=PQ^2 =a^2 (1−μ−1)^2 +a^2 (1−μ−1+λ)^2 +a^2 (1−λ)^2 =a^2 {μ^2 +(λ−μ)^2 +(1−λ)^2 } =a^2 {2λ^2 +2μ^2 −2λμ−2λ+1} (∂D/∂λ)=0⇒4λ−2μ−2=0⇒2λ−μ=1 (∂D/∂μ)=0⇒4μ−2λ=0⇒2μ−λ=0 ⇒λ=(2/3), μ=(1/3) D_(min) =a^2 ((1/9)+(1/9)+(1/9))=(a^2 /3) ⇒PQ_(min) =(√D_(min) )=(a/(√3))$
	$A (a, a, 0)$ $B (a, 0, a)$ $e q n . o f l i n e A B :$ $\frac{x - a}{a - a} = \frac{y - a}{0 - a} = \frac{z - 0}{a - 0} = λ$ $o r$ ${\begin{cases} x = a \\ y = a (1 - λ) \\ z = a λ \end{cases}$ $C (a, a, a)$ $D (0, 0, a)$ $e q n . o f l i n e C D :$ $\frac{x - a}{0 - a} = \frac{y - a}{0 - a} = \frac{z - a}{a - a} = μ$ $o r$ ${\begin{cases} x = a (1 - μ) \\ y = a (1 - μ) \\ z = a \end{cases}$ $D = {P Q}^{2} = a^{2} {(1 - μ - 1)}^{2} + a^{2} {(1 - μ - 1 + λ)}^{2} + a^{2} {(1 - λ)}^{2}$ $= a^{2} {μ^{2} + {(λ - μ)}^{2} + {(1 - λ)}^{2}}$ $= a^{2} {2 λ^{2} + 2 μ^{2} - 2 λ μ - 2 λ + 1}$ $\frac{\partial D}{\partial λ} = 0 \Rightarrow 4 λ - 2 μ - 2 = 0 \Rightarrow 2 λ - μ = 1$ $\frac{\partial D}{\partial μ} = 0 \Rightarrow 4 μ - 2 λ = 0 \Rightarrow 2 μ - λ = 0$ $\Rightarrow λ = \frac{2}{3}, μ = \frac{1}{3}$ $D_{m i n} = a^{2} (\frac{1}{9} + \frac{1}{9} + \frac{1}{9}) = \frac{a^{2}}{3}$ $\Rightarrow {P Q}_{m i n} = \sqrt{D_{m i n}} = \frac{a}{\sqrt{3}}$
	Commented by ajfour last updated on 18/May/19

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