Find the equation of the line in vector and in cartesion form that passes through the point with position vector $2^i -^j + 4^k$ and is in the direction $^i + 2^j -^k$ .

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Solution

$V e c t o r e q u a t i o n o f a l i n e i s :$
$¯ ¯ ¯ r = ¯ ¯ ¯ a + λ ¯ ¯ b$
$w h e r e ¯ ¯ ¯ r i s a p o i n t o n t h e l i n e, ¯ ¯ ¯ a i s a k n o w n$
$p o i n t l y i n g o n l i n e, a n d ¯ ¯ b i s t h e l i n e t o w h i c h$
$t h e g i v e n l i n e i s p a r a l l e l,$
$⟹ ¯ ¯ ¯ r = (2 \land i - \land j + 4 \land k) + λ (\land i + 2 \land j - \land k)$
$F o r c a r t e s i a n f o r m :$
$∵ T h e g i v e n l i n e p a s s e s t h r o u g h (2, - 1, 4) a n d d i r e c t i o n .$
$n o . i s a = 1, b = 2, c = - 1$
$E q u a t i o n i s : \frac{x - x 1}{a} = \frac{y = y 1}{b} = \frac{z - z 1}{c}$
$C a r t e s i a n e q u a t i o n,$
$⟹ \frac{x - 2}{1} = \frac{y - (- 1)}{2} = \frac{z - 4}{- 1}$
$⟹ \frac{x - 2}{1} = \frac{y + 1}{2} = \frac{z - 4}{- 1}$

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Similar questions

2^i -^j + 4^k

^i + 2^j -^k

2^i -^j +^k

-^i + 4^j +^k

^i + 2^j +^k

2^i +^j -^k

-^i + 3^j + 4^k

⊥

¯ ¯ ¯ r . (^i - 2^j + 4^k) = 10

2^i +^j - 3^k

^i +^j +^k

^i + 2^j -^k

^i +^j - 2^k, 2^i -^j +^k

^i + 2^j +^k

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

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