SATHEE: Vectors 5 Question 16

Vectors 5 Question 16

16. Find all values of $λ$ , such that $x, y, z \neq (0, 0, 0)$ and $(\hat{i} + \hat{j} + 3 \hat{k}) x + (3 \hat{i} - 3 \hat{j} + \hat{k}) y$ $+ (- 4 \hat{i} + 5 \hat{j}) z = λ (\hat{i} x + \hat{j} y + \hat{k} z)$ , where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the coordinate axes.

(1982, 2M)

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Solution:

Since, $(\hat{i} + \hat{j} + 3 \hat{k}) x + (3 \hat{i} - 3 \hat{j} + \hat{k}) y + (- 4 \hat{i} + 5 \hat{j}) z$

$= λ (\hat{i} x + \hat{j} y + \hat{k} z)$

$\Rightarrow x + 3 y - 4 z = λ x, x - 3 y + 5 z = λ y, 3 x + y + 0 z = λ z$

$\Rightarrow (1 - λ) x + 3 y - 4 z = 0, x - (3 + λ) y + 5 z = 0$ ,

$3 x + y - λ z = 0$

Since, $(x, y, z) \neq (0, 0, 0)$

$∴$ Non-trivial solution.

$\begin{aligned} \Rightarrow Δ = 0 \\ \Rightarrow | \begin{array}{ccc} 1 - λ & 3 & - 4 \\ 1 & - (3 + λ) & 5 \\ 3 & 1 & - λ \end{array} | = 0 \\ \Rightarrow (1 - λ) (3 λ + λ^{2} - 5) - 3 (- λ - 15) - 4 (1 + 9 + 3 λ) = 0 \\ \Rightarrow - λ^{3} - 2 λ^{2} - λ = 0 \Rightarrow λ (λ + 1)^{2} = 0 \\ ∴ λ = 0, - 1 \end{aligned}$

Vectors 5 Question 16

16. Find all values of $λ$ , such that $x, y, z \neq (0, 0, 0)$ and $(\hat{i} + \hat{j} + 3 \hat{k}) x + (3 \hat{i} - 3 \hat{j} + \hat{k}) y$ $+ (- 4 \hat{i} + 5 \hat{j}) z = λ (\hat{i} x + \hat{j} y + \hat{k} z)$ , where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the coordinate axes.

Solution:

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Vectors 5 Question 16

16. Find all values of λ, such that x,y,z≠(0,0,0) and (i^+j^+3k^)x+(3i^−3j^+k^)y +(−4i^+5j^)z=λ(i^x+j^y+k^z), where i^,j^,k^ are unit vectors along the coordinate axes.

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16. Find all values of $λ$ , such that $x, y, z \neq (0, 0, 0)$ and $(\hat{i} + \hat{j} + 3 \hat{k}) x + (3 \hat{i} - 3 \hat{j} + \hat{k}) y$ $+ (- 4 \hat{i} + 5 \hat{j}) z = λ (\hat{i} x + \hat{j} y + \hat{k} z)$ , where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the coordinate axes.