The vector equation of the line passing through 3-5-7k- and perpendicular to the plane 3x-4y-5z-8 iswhere - is a parameter

Given plane is nbsp; 3x 4y+5z=8 or (3î 4ĵ+5k̂)·(xî+yĵ+zk̂)=8 This shows that n=(3î 4ĵ+5k̂) is normal to the given plane. ∴ The required line is parallel to (3 ...

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Standard XII

Mathematics

Equation of a Plane : Point Normal Form

The vector eq...

Question

The vector equation of the line passing through $3^i - 5^j + 7^k$ and perpendicular to the plane $3 x - 4 y + 5 z = 8$ is
(where $λ$ is a parameter)

3^i - 5^j + 7^k + λ (3^i + 5^j - 4^k)

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3^i - 5^j + 7^k + λ (2^i -^j + 3^k)

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3^i - 5^j + 7^k + λ (3^i - 4^j + 5^k)

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3^i - 5^j + 7^k + λ (2^i + 3^j + 5^k)

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Solution

The correct option is C $3^i - 5^j + 7^k + λ (3^i - 4^j + 5^k)$
Given plane is $3 x - 4 y + 5 z = 8$ or $(3^i - 4^j + 5^k) \cdot (x^i + y^j + z^k) = 8$
This shows that $\to n = (3^i - 4^j + 5^k)$ is normal to the given plane.
$∴$ The required line is parallel to $(3^i - 4^j + 5^k)$
Since the required line passes through $3^i - 5^j + 7^k$
$∴$ It's equation is given by
$3^i - 5^j + 7^k + λ (3^i - 4^j + 5^k)$
Where $λ$ is a parameter.

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Similar questions

Q. The vector equation of a line passing through the point

¯ i + 2 ¯ j + 3 ¯ ¯ ¯ k

and perpendicular to the vectors

¯ i + ¯ j + ¯ ¯ ¯ k

and

2 ¯ i - ¯ j + ¯ ¯ ¯ k

Q. Equation of the line passing through

i + j - 3 k

and perpendicular to the plane

2 x - 4 y + 3 x + 5 = 0

Q. If the volume of the tetrahedron whose verticles are with position vectors

^i - 6^j + 10^k, -^i - 3^j + 7^k, 5^i -^j + λ^k

and

7^i - 4^j + 7^k

is 11 cubic units, then find the value of

λ

Q. A plane passing through the point(-1,1,1) is parallel to the vector

2 ˆ i + 3 ˆ j - 7 ˆ k

and line

r = (ˆ i - 2 ˆ j - ˆ k) + λ (3 ˆ i - 8 ˆ j + 2 ˆ k)

. Find the vector equation of the plane.

Q. Find

λ

if the vectors

\to a =^i + 3^j +^k \to b = 2^i -^j -^k \to c = λ^i + 7^j + 3^k

are coplanar.

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The vector equation of the line passing through 3^i−5^j+7^k and perpendicular to the plane 3x−4y+5z=8 is (where λ is a parameter)

The vector equation of the line passing through $3^i - 5^j + 7^k$ and perpendicular to the plane $3 x - 4 y + 5 z = 8$ is
(where $λ$ is a parameter)