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

Let A, B, C and D be the given points whose position vectors are #160; #160;j #160;k, 4 #160;i + 5 #160;j + λ #160;k, 3 #160;i + 9 # ...

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Standard XII

Mathematics

Skew Lines

Find the valu...

Question

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.

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Solution

Let A, B, C and D be the given points whose position vectors are
$- ˆ j - ˆ k, 4 ˆ i + 5 ˆ j + λ ˆ k, 3 ˆ i + 9 ˆ j + 4 ˆ k$ and $- 4 ˆ i + 4 ˆ j + 4 ˆ k$ respectively.

$A \to B = 4 ˆ i + 5 ˆ j + λ ˆ k - (- ˆ j - ˆ k) = 4 ˆ i + 6 ˆ j + (λ + 1) ˆ k$

$A \to C = 3 ˆ i + 9 ˆ j + 4 ˆ k - (- ˆ j - ˆ k) = 3 ˆ i + 10 ˆ j + 5 ˆ k$

$A \to D - 4 ˆ i + 4 ˆ j + 4 ˆ k - (- ˆ j - ˆ k) = - 4 ˆ i + 5 ˆ j + 5 ˆ k$

Since the points $A, B, C$ and $D$ are coplanar, vectors $A \to B, A \to C$ and $A \to D$ are coplanar.
$\Rightarrow [- - \to A B - - \to A C - - \to A D] = 0$
$\Rightarrow 4 (50 - 25) - 3 (30 - 5 - 5 λ) - 4 (30 - 10 - 10 λ) = 0$
$\Rightarrow 100 - 75 + 15 λ - 80 + 40 λ = 0$
$\Rightarrow - 55 + 55 λ = 0$
$\Rightarrow - 1 + λ = 0$
$\Rightarrow λ = 1$

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Similar questions

The position vectors of four [point A, B, C and D are $ˆ i + ˆ j + ˆ k, 2 ˆ i + 5 ˆ j, 3 ˆ i + 2 ˆ j - 3 ˆ k, ˆ i - 6 ˆ j - ˆ k$ then select the correct relation:

Q. If vectors

\to a = ˆ i + 2 ˆ j - ˆ k, \to b = 2 ˆ i - ˆ j + ˆ k

and

\to c = λ ˆ i + ˆ j + 2 ˆ k

are coplanar, then find the value of

(λ - 4)

Q. The cross product of vectors A =

(ˆ i + ˆ j)

and B =

(ˆ i + ˆ k)

Q. Show that the points A, B, C with position vectors

2 ˆ i - ˆ j + ˆ k, ˆ i - 3 ˆ j - 5 ˆ k

and

3 ˆ i - 4 ˆ j - 4 ˆ k

respectively, are the vertices of a right angled triangle. Hence find the area of the triangle.

Q. The points whose position vectors are

5 ˆ i + 5 ˆ k, 2 ˆ i + ˆ j + 3 ˆ k

and

- 4 ˆ i + 3 ˆ j - ˆ k

are collinear.

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Find the value of λ for which the four points A,B,C,D with position vectors −ˆj−ˆk;4ˆi+5ˆj+λˆk;3ˆi+9ˆj+4ˆk and −4ˆi+4ˆj+4ˆk are coplanar.

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.