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Câu hỏi: Cho hình lập phương $ABCD.{A}'{B}'{C}'{D}'$ có cạnh $a$ (tham khảo hình bên). Khoảng cách giữa hai đường thẳng $AC$ và $D{C}'$ bằng
A. $a.$
B. $\dfrac{a\sqrt{3}}{2}.$
C. $\dfrac{a\sqrt{3}}{3}.$
D. $\dfrac{a}{3}.$
Ta có $AC\text{//}\left( {A}'{C}'D \right)\Rightarrow d\left( AC,D{C}' \right)=d\left( AC,\left( {A}'{C}'D \right) \right)=d\left( C,\left( {A}'{C}'D \right) \right)=d\left( {D}',\left( {A}'{C}'D \right) \right)$
Gọi $I={A}'{C}'\cap {B}'{D}'$.
Ta có $\left( {A}'{C}'D \right)\bot \left( D{D}'I \right)$, kẻ ${D}'H\bot DI\Rightarrow {D}'H\bot \left( {A}'{C}'D \right)$
$\Rightarrow d\left( {D}',\left( {A}'{C}'D \right) \right)={D}'H=\dfrac{{D}'I.D{D}'}{DI}=\dfrac{a\sqrt{3}}{3}$
Suy ra $d\left( AC,D{C}' \right)=\dfrac{a\sqrt{3}}{3}$.
A. $a.$
B. $\dfrac{a\sqrt{3}}{2}.$
C. $\dfrac{a\sqrt{3}}{3}.$
D. $\dfrac{a}{3}.$
Ta có $AC\text{//}\left( {A}'{C}'D \right)\Rightarrow d\left( AC,D{C}' \right)=d\left( AC,\left( {A}'{C}'D \right) \right)=d\left( C,\left( {A}'{C}'D \right) \right)=d\left( {D}',\left( {A}'{C}'D \right) \right)$
Gọi $I={A}'{C}'\cap {B}'{D}'$.
Ta có $\left( {A}'{C}'D \right)\bot \left( D{D}'I \right)$, kẻ ${D}'H\bot DI\Rightarrow {D}'H\bot \left( {A}'{C}'D \right)$
$\Rightarrow d\left( {D}',\left( {A}'{C}'D \right) \right)={D}'H=\dfrac{{D}'I.D{D}'}{DI}=\dfrac{a\sqrt{3}}{3}$
Suy ra $d\left( AC,D{C}' \right)=\dfrac{a\sqrt{3}}{3}$.
Đáp án C.