solve

$C o n s i d e r t h e l i n e s \bar{r} = (i + 2 j - 2 k) + λ (i + 2 j) a n d \bar{r} = (i + 2 j - 2 k) + μ (2 j - k)$
a) Find the angle between the lines.
b) Find a vector perpendicular to both the lines.
c) Find the equation of the line passing through the point of intersection of lines and perpendicular to both the lines.

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Solution

$L 1 : \overset{⇀}{r} = \overset{⏜}{i} + 2 \overset{⏜}{j} - 2 \overset{⏜}{k} + λ (\overset{⏜}{i} + 2 \overset{⏜}{j}) L 2 : \overset{⇀}{r} = \overset{⏜}{i} + 2 \overset{⏜}{j} - 2 \overset{⏜}{k} + μ (2 \overset{⏜}{j} - \overset{⏜}{k}) T h e t e r m i n s i d e b r a c k e t s a lo n g s i d e λ a n d μ r e p r e s e n t v e c t o r s p a r a l l e l t o L 1 a n d L 2 r e s p e c t i v e l y v e c t o r p a r a l l e l t o L 1, \overset{⇀}{b_{1}} = \overset{⏜}{i} + 2 \overset{⏜}{j} v e c t o r p a r a l l e l t o L 2, \overset{⇀}{b_{2}} = 2 \overset{⏜}{j} - \overset{⏜}{k} A n g l e b e t w e e n l i n e s C o s θ = |\frac{\overset{⇀}{b_{1}} . \overset{⇀}{b_{2}}}{|\overset{⇀}{b_{1}}| |\overset{⇀}{b_{1}}|}| = \frac{4}{\sqrt{5} \sqrt{5}} = \frac{4}{5} θ = \cos^{- 1} \frac{4}{5} V e c t o r p e r p e n d i c u l a r b o t h t h e l i n e s = \overset{⇀}{b_{1}} \times \overset{⇀}{b_{2}} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 0 \\ 0 & 2 & - 1 \end{matrix}| = - 2 \overset{⏜}{i} + \overset{⏜}{j} + 2 \overset{⏜}{k} O b s e r v e e q u a t i o n o f L 1 a n d L 2 t h e v e c t o r i n d e p e n d e n t o f λ a n d μ r e p r e s e n t t h e p o s i t i o n v e c t o r o f p o i n t t h r o u g h w h i c h L 1 a n d L 2 p a s s e s S i n c e t h e y b o t h p a s s t h r o u g h \overset{⏜}{i} + 2 \overset{⏜}{j} - 2 \overset{⏜}{k} t h e r e f o r e t h e y i n t e r s e c t a t \overset{⏜}{i} + 2 \overset{⏜}{j} - 2 \overset{⏜}{k} E q u a t i o n o f l i n e L p a s \sin g t h r o u g h \overset{⏜}{i} + 2 \overset{⏜}{j} - 2 \overset{⏜}{k} a n d p a r a l l e l t o - 2 \overset{⏜}{i} + \overset{⏜}{j} + 2 \overset{⏜}{k} \overset{⇀}{r} = \overset{⏜}{i} + 2 \overset{⏜}{j} - 2 \overset{⏜}{k} + α (- 2 \overset{⏜}{i} + \overset{⏜}{j} + 2 \overset{⏜}{k})$

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(3, - 1, 2)

$¯ ¯ ¯ r = ¯ i + ¯ j - ¯ ¯ ¯ k + λ (2 ¯ i - 2 ¯ j + 2 ¯ ¯ ¯ k)$ and $¯ ¯ ¯ r = 2 ¯ i + ¯ j - 3 ¯ ¯ ¯ k + μ (¯ i - 2 ¯ j + 2 ¯ ¯ ¯ k)$

\to r = 3 i + 2 j - 4 k + λ (i + 2 j + 2 k)

\to r = (5 j - 2 k) + μ (3 i + 2 j + 6 k)

\to r = (6 ˆ i + 2 ˆ j + 2 ˆ k) + λ (ˆ i - 2 ˆ j + 2 ˆ k)

\to r = (- 4 ˆ i - ˆ k) + μ (3 ˆ i - 2 ˆ j - 2 ˆ k)

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(2, - 1, 3)

r = i + j - k + λ (2 i - 2 j + k)

r = 2 i - j - 3 k + μ (i + 2 j + 2 k)

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