SATHEE: Vectors 3 Question 30

Vectors 3 Question 30

30. The position vectors of the points $A, B, C$ and $D$ are $3 \hat{i} - 2 \hat{j} - \hat{k}, 2 \hat{i} + 3 \hat{j} - 4 \hat{k}, - \hat{i} + \hat{j} + 2 \hat{k}$ and $4 \hat{i} + 5 \hat{j} + λ \hat{k}$ , respectively. If the points $A, B, C$ and $D$ lie on a plane, find the value of $λ$ .

$(1986, 2 \frac{1}{2}$ M)

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Solution:

Here,

$\vec{A B} = - \hat{i} + 5 \hat{j} - 3 \hat{k}$

$\vec{A C} = - 4 \hat{i} + 3 \hat{j} + 3 \hat{k}$ and $\vec{A D} = \hat{i} + 7 \hat{j} + (λ + 1) \hat{k}$

We know that, $A, B, C, D$ lie in a plane if $\vec{A B}, \vec{A C}, \vec{A D}$ are coplanar i.e.

$\begin{array}{ccc} [\vec{A B} \vec{A C} \vec{A D}] = 0 \Rightarrow | \begin{array}{ccc} - 1 & 5 & - 3 \\ - 4 & 3 & 3 \\ 1 & 7 & λ + 1 \end{array} | = 0 \\ \Rightarrow & - 1 (3 λ + 3 - 21) - 5 (- 4 λ - 4 - 3) - 3 (- 28 - 3) = 0 \\ \Rightarrow & - 17 λ + 146 = 0 \\ ∴ & λ = \frac{146}{17} \end{array}$

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Vectors 3 Question 30

30. The position vectors of the points A,B,C and D are 3i^−2j^−k^,2i^+3j^−4k^,−i^+j^+2k^ and 4i^+5j^+λk^, respectively. If the points A,B,C and D lie on a plane, find the value of λ.

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30. The position vectors of the points $A, B, C$ and $D$ are $3 \hat{i} - 2 \hat{j} - \hat{k}, 2 \hat{i} + 3 \hat{j} - 4 \hat{k}, - \hat{i} + \hat{j} + 2 \hat{k}$ and $4 \hat{i} + 5 \hat{j} + λ \hat{k}$ , respectively. If the points $A, B, C$ and $D$ lie on a plane, find the value of $λ$ .