Find the value of - for which the four points with position vectors -j-k- 4 i-5 j- k- 3 i-9 j-4 k- and -4 i -4 j-4 k - are coplanar-

Let #160;A, #160;B, #160;C #160;and #160;D #160;be #160;the #160;given #160;points. #160;Then,AB #8594; #160;=(4i^+5j^+ #955;k^) ( # ...

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Standard XII

Mathematics

Latus Rectum

Find the valu...

Question

Find the value of λ for which the four points with position vectors $- \hat{j} - \hat{k}, 4 \hat{i} + 5 \hat{j} + λ \hat{k}, 3 \hat{i} + 9 \hat{j} + 4 \hat{k} and - 4 \hat{i} + 4 \hat{j} + 4 \hat{k}$ are coplanar.

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Solution

$Let A, B, C and D be the given points . Then, \vec{A B} = (4 \hat{i} + 5 \hat{j} + λ \hat{k}) - (0 \hat{i} - \overset{⏜}{j} - \overset{⏜}{k}) = 4 \hat{i} + 6 \hat{j} + (λ + 1) \hat{k} \vec{A C} = (3 \hat{i} + 9 \hat{j} + 4 \hat{k}) - (0 \hat{i} - \hat{j} - \hat{k}) = 3 \hat{i} + 10 \hat{j} + 5 \hat{k} \vec{A D} = (- 4 \hat{i} + 4 \hat{j} + 4 \hat{k}) - (0 \hat{i} - \hat{j} - \hat{k}) = - 4 \hat{i} + 5 \hat{j} + 5 \hat{k} The given points are coplanar iff vector s \vec{A B}, \vec{A C}, \vec{A D} are coplanar . Now, \vec{A B}, \vec{A C}, \vec{A D} are coplanar . \Rightarrow [\begin{matrix} \vec{A B} & \vec{A C} & \vec{A D} \end{matrix}] = 0 \Rightarrow |\begin{matrix} 4 & 6 & (λ + 1) \\ 3 & 10 & 5 \\ - 4 & 5 & 5 \end{matrix}| = 0 \Rightarrow 4 (50 - 25) - 6 (15 + 20) + (λ + 1) (15 + 40) = 0 \Rightarrow 100 - 210 + 55 λ + 55 = 0 \Rightarrow 55 λ = 55 \Rightarrow λ = 1$

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Similar questions

Q. If the points whose position vectors are

3 ¯ i - 2 ¯ j - ¯ ¯ ¯ k, 2 ¯ i + 3 ¯ j - 4 ¯ ¯ ¯ k, - ¯ i + ¯ j + 2 ¯ ¯ ¯ k

and

4 ¯ i + 5 ¯ j + λ ¯ ¯ ¯ k

are coplanar then show that

λ = \frac{- 146}{m}

.Find

m

Q. Find the value of

λ

for which the four points

A, B, C, D

with position vectors

- ˆ j - ˆ k; 4 ˆ i + 5 ˆ j + λ ˆ k; 3 ˆ i + 9 ˆ j + 4 ˆ k

and

- 4 ˆ i + 4 ˆ j + 4 ˆ k

are coplanar.

P, Q, R

and

S

are four points with the position vectors

3 ¯ i - 4 ¯ j + 5 ¯ ¯ ¯ k, 4 ¯ ¯ ¯ k, - 4 ¯ i + 5 ¯ j + ¯ ¯ ¯ k

and

- 3 ¯ i + 4 ¯ j + 3 ¯ ¯ ¯ k

respectively. Then the line

P Q

meets the line

R S

at the point.

P,Q,R and S are four points with the position vectors, $3 \to i - 4 \to j + 5 \to k, 4 \to k, - 4 \to i + 5 \to j + \to k and - 3 \to i + 4 \to j + 3 \to k$ respectively. Then the line PQ meets the line RS at the point

Q. For what value of λ are the vectors

\vec{a} and \vec{b}

perpendicular to each other if
(i)

\vec{a} = λ \hat{i} + 2 \hat{j} + \hat{k} and \vec{b} = 4 \hat{i} - 9 \hat{j} + 2 \hat{k}

(ii)

\vec{a} = λ \hat{i} + 2 \hat{j} + \hat{k} and \vec{b} = 5 \hat{i} - 9 \hat{j} + 2 \hat{k}

(iii)

\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} and \vec{b} = 3 \hat{i} + 2 \hat{j} - λ \hat{k}

(iv)

\vec{a} = λ \hat{i} + 3 \hat{j} + 2 \hat{k} and \vec{b} = \hat{i} - \hat{j} + 3 \hat{k}

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Find the value of λ for which the four points with position vectors -j^-k^, 4i^+5j^+λk^, 3i^+9j^+4k^ and -4i^+4j^+4k^ are coplanar.

Find the value of λ for which the four points with position vectors $- \hat{j} - \hat{k}, 4 \hat{i} + 5 \hat{j} + λ \hat{k}, 3 \hat{i} + 9 \hat{j} + 4 \hat{k} and - 4 \hat{i} + 4 \hat{j} + 4 \hat{k}$ are coplanar.