Find the value of - if four points with position vectors 3- - 6 - - 9 k- 2 - - 3k- 2 -3- - k- and 4- - 6 - - -k- are coplanar-

Let A, B, C and D be the given points. Then, AB = (î + 2ĵ + 3k̂ ) (3 î + 6ĵ + 9k̂ ) = 2 î 4 ĵ 6 k̂ AC = (2î +3 ĵ+ k̂ ) (3 î + 6 ĵ + 9 k̂ ) = î 3ĵ 8k ...

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Standard IX

Mathematics

Bisector of Angle between Two Vectors

Find the valu...

Question

Find the value of $λ$ , if four points with position vectors $3^i + 6^j + 9^k,^i + 2^j + 3^k, 2^i + 3^j +^k$ and $4^i + 6^j + λ^k$ are coplanar.

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Solution

Let $A, B, C and D$ be the given points. Then, $- - \to A B = (^i + 2^j + 3^k) - (3^i + 6^j + 9^k) = - 2^i - 4^j - 6^k - - \to A C = (2^i + 3^j +^k) - (3^i + 6^j + 9^k) = -^i - 3^j - 8^k - - \to A D = (4^i + 6^j + λ^k) - (3^i + 6^j + 9^k) =^i + 0^j + (λ - 9)^k$
Given points are coplanar if vectors
$- - \to A B, - - \to A C, - - \to A D$ are coplanar
$[- - \to A B, - - \to A C, - - \to A D] = 0 \Rightarrow - 2 (λ - 9) (- 3) + 4 (- λ + 9 + 8) - 6 (3) = 0 \Rightarrow 6 λ - 54 - 4 λ + 68 - 18 = 0 \Rightarrow 2 λ - 4 = 0 \Rightarrow λ = 2$

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Similar questions

Q. Find the value of

λ

, if four points with position vectors

3^i + 6^j + 9^k,^i + 2^j + 3^k, 2^i + 3^j +^k

and

4^i + 6^j + λ^k

are coplanar.

Find the value of $λ$ ,if four points with position vectors $3^i + 6^j + 9^k,^i + 2^j + 3^k, 2^i + 3^j +^k$ and $4^i + 6^j + λ^k$ are coplanar.

Q. The unit vector which is orthogonal to the vector

3^i + 2^j + 6^k

and is coplanar with the vectors

2^i +^j +^k a n d^i -^j +^k

Q. A unit vector perpendicular to the plane of

a = 2^i - 6^j - 3^k, b = 4^i + 3^j -^k

Q. If two vectors

2^i + 3^j + 3^k

and

- 4^i - 6^j + λ^k

are parallel to each other then value of

λ

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Find the value of λ, if four points with position vectors 3^i+6^j+9^k,^i+2^j+3^k,2^i+3^j+^k and 4^i+6^j+λ^k are coplanar.

Find the value of $λ$ , if four points with position vectors $3^i + 6^j + 9^k,^i + 2^j + 3^k, 2^i + 3^j +^k$ and $4^i + 6^j + λ^k$ are coplanar.