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Equation of a Plane Passing through a Point and Parallel to the Two Given Vectors

The vector eq...

Question

The vector equation of line passing through $(2, 1, - 4)$ and parallel to a vector $¯ i + ¯ j - 2 ¯ ¯ ¯ k$ is

¯ ¯ ¯ r = 2 ¯ i - ¯ j - 4 ¯ ¯ ¯ k + λ (¯ i + ¯ j - 2 ¯ ¯ ¯ k)

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¯ ¯ ¯ r = 2 ¯ i - ¯ j + 4 ¯ ¯ ¯ k - λ (¯ i - ¯ j - 2 ¯ ¯ ¯ k)

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¯ ¯ ¯ r = 2 ¯ i - ¯ j + 4 ¯ ¯ ¯ k + λ (¯ i + ¯ j + 2 ¯ ¯ ¯ k)

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¯ ¯ ¯ r = 2 ¯ i + ¯ j - 4 ¯ ¯ ¯ k + λ (¯ i + ¯ j - 2 ¯ ¯ ¯ k)

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Solution

The correct option is D $¯ ¯ ¯ r = 2 ¯ i + ¯ j - 4 ¯ ¯ ¯ k + λ (¯ i + ¯ j - 2 ¯ ¯ ¯ k)$
Given line is passing through $P (2, 1, - 4)$
and parallel to $^i +^j - 2^k$
so normal vector of given line is
$\to n =^i +^j - 2^k$
eq of line through point $P (2, 1, - 4)$
$\to r = \to a + λ \to n$
$\to r = 2^i +^j - 4^k + λ (^i +^j - 2^k)$

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