x − λ 1 = y − 1 2 1 2 = z − 1 2 {{x - \lambda } \over 1} = {{y - {1 \over 2}} \over {{1 \over 2}}} = {z \over { - {1 \over 2}}} 1 x − λ = 2 1 y − 2 1 = − 2 1 z
x − λ 2 = y − 1 2 1 = 2 − 1 {{x - \lambda } \over 2} = {{y - {1 \over 2}} \over 1} = {2 \over { - 1}} 2 x − λ = 1 y − 2 1 = − 1 2
Point on line = ( λ , 1 2 , 0 ) \left( {\lambda ,{1 \over 2},0} \right) ( λ , 2 1 , 0 )
x 1 = y + 2 λ 1 = z − λ 1 {x \over 1} = {{y + 2\lambda } \over 1} = {{z - \lambda } \over 1} 1 x = 1 y + 2 λ = 1 z − λ
Point on line = ( 0 , − 2 λ , λ ) (0, - 2\lambda ,\lambda ) ( 0 , − 2 λ , λ )
Distance between skew lines d = ∣ ( a ⃗ 2 − a ⃗ 1 ) ⋅ ( b ⃗ 1 × b ⃗ 2 ) ∣ b ⃗ 1 × b ⃗ 2 ∣ ∣ d = \left| \frac{(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|} \right| d = ∣ b 1 × b 2 ∣ ( a 2 − a 1 ) ⋅ ( b 1 × b 2 )
∣ λ 1 2 + 2 λ − λ 2 1 − 1 1 1 1 ∣ = v ⃗ ⋅ ∣ i ^ j ^ k ^ 2 1 − 1 1 1 1 ∣ \\ \left| \begin{matrix} \lambda & \frac{1}{2} + 2\lambda & -\lambda \\ 2 & 1 & -1 \\ 1 & 1 & 1 \end{matrix} \right| = \vec{v} \cdot \left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -1 \\ 1 & 1 & 1 \end{matrix} \right| λ 2 1 2 1 + 2 λ 1 1 − λ − 1 1 = v ⋅ i ^ 2 1 j ^ 1 1 k ^ − 1 1
= ∣ − 5 λ − 3 2 ∣ 14 = 7 2 2 = {{\left| { - 5\lambda - {3 \over 2}} \right|} \over {\sqrt {14} }} = {{\sqrt 7 } \over {2\sqrt 2 }} = 14 ∣ − 5 λ − 2 3 ∣ = 2 2 7
= ∣ 10 λ + 3 ∣ = 7 ⇒ λ = − 1 = |10\lambda + 3| = 7 \Rightarrow \lambda = - 1 = ∣10 λ + 3∣ = 7 ⇒ λ = − 1
⇒ ∣ λ ∣ = 1 \Rightarrow |\lambda | = 1 ⇒ ∣ λ ∣ = 1
