Show that the four points A- B- C and D with position vectors 4-5-k- -k- 3-9-4k- and 4-k- respectively- are coplanar-

We know that four points A,B,C,D are coplanar if the three vectors A⃗B⃗,A⃗C⃗,A⃗D⃗ are coplanar.That is, [A⃗B⃗ A⃗C⃗ A⃗D⃗]=0 A⃗B⃗=( ĵ k̂) ...

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Byju's Answer

Standard XII

Mathematics

Definition of Vector

Show that the...

Question

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.

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Solution

We know that four points $A, B, C, D$ are coplanar if the three vectors $\to A B, \to A C, \to A D$ are coplanar.

That is, $[\to A B \to A C \to A D] = 0$

$\to A B = (-^j -^k) - (4^i + 5^j +^k) = - 4^i - 6^j - 2^k$

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

$\to A D = (- 4^i + 4^j + 4^k) - (4^i + 5^j +^k) = - 8^i -^j + 3^k$

$[\to A B \to A C \to A D] = ∣ ∣ ∣ ∣ \begin{matrix} - 4 & - 6 & - 2 - 1 & 4 & 3 - 8 & - 1 & 3 \end{matrix} ∣ ∣ ∣ ∣$

$= - 4 (12 + 3) + 6 (- 3 + 24) - 2 (1 + 32)$

$= - 4 (15) + 6 (21) - 2 (33)$

$= - 60 + 126 - 66$

$= - 126 + 126$

$= 0$

$∴ [\to A B \to A C \to A D] = 0$

Hence the four points $A, B, C, D$ are coplanar.

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Similar questions

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 $\to b + \to c$ .