Find the value of - so that the four points A- B- C and D with position vectors- 4- - 4- - k- - - - k-3- - - - 4k- and - 4- - 4- - 4k- respectively are coplanar-

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Condition for Coplanarity of Four Points

Find the valu...

Question

Find the value of $λ$ so that the four points $A, B, C$ and $D$ with position vectors, $4^i + 4^j +^k, -^j -^k, 3^i + λ^j + 4^k$ and $- 4^i + 4^j + 4^k$ respectively are coplanar.

Open in App

Solution

$\begin{matrix} A (4^i + 4^j +^k) B (-^j -^k) C (3^i + λ^j + 4^k) D (- 4^i + 4^j + 4^k) - - \to A B = - 4^i + 5^j - 2^k - - \to A C = -^i + (λ - 4)^j + 3^k - - \to A D = - 8^i + 3^k [- - \to A B - - \to A C - - \to A D] = 0 (A, B, C, D a r e c o p l a n a r) ∣ ∣ ∣ ∣ \begin{matrix} - 4 & - 5 & - 2 - 1 & (λ - 4) & 3 - 8 & 0 & 3 \end{matrix} ∣ ∣ ∣ ∣ = 0 - 4 [3 (λ - 4)] + 5 [- 3 + 24] - 2 [0 + 8 (λ - 4)] - 12 (λ - 4) + 5 (21) - 16 (λ - 4) = 0 - 28 (λ - 4) + 105 = 0 - 4 (λ - 4) + 15 = 0 31 = 4 λ H e n c e, λ = \frac{31}{4} \end{matrix}$

Suggest Corrections

Similar questions

Q. If the points with position vectors

- (^j +^k), 4^i + 5^j + λ^k, 3^i + 9^j + 4^k

and

- 4^i + 4^j + 4^k

are coplanar, then the value of

λ

Q. If the points with position vectors

- (^j +^k), 4^i + 5^j + λ^k, 3^i + 9^j + 4^k

and

- 4^i + 4^j + 4^k

are coplanar, then the value of

λ

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

4^i + 5^j +^k, -^j -^k, 3^i + 9^j + 4^k

and

4 (-^i +^j +^k)

respectively, are coplanar.

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

4^i + 5^j +^k, -^j -^k, 3^i + 9^j + 4^k and 4 (-^i +^j +^k)

respectively are coplanar.

Show that the four points A, B, C and D with position vectors $4^i + 5^j +^k, -^j -^k, 3^i + 9^j + 4^k$ and $4 (-^i +^j +^k)$ respectively are coplannar.

The scalar product of the vector $\to a =^i +^j +^k$ with a unit vector along the sum of vector $\to b = 2^i + 4^j + 5^k$ and $\to c = λ^i + 2^j + 3^k$ is equal to one. Find the value of $λ$ and hence find the unit vector along $\to b + \to c$ .

Join BYJU'S Learning Program

Find the value of λ so that the four points A,B,C and D with position vectors, 4^i+4^j+^k,−^j−^k,3^i+λ^j+4^k and −4^i+4^j+4^k respectively are coplanar.

Find the value of $λ$ so that the four points $A, B, C$ and $D$ with position vectors, $4^i + 4^j +^k, -^j -^k, 3^i + λ^j + 4^k$ and $- 4^i + 4^j + 4^k$ respectively are coplanar.