SATHEE: Vectors 3 Question 1

Vectors 3 Question 1

1. The sum of the distinct real values of $μ$ , for which the vectors, $μ \hat{i} + \hat{j} + \hat{k}, \hat{i} + μ \hat{j} + \hat{k}, \hat{i} + \hat{j} + μ \hat{k}$ are coplanar, is

(2019 Main, 12 Jan I)

(a) 2

(b) 0

(d) -1

Show Answer

Answer:

Correct Answer: 1. (d)

Solution:

Given vectors, $μ \hat{i} + \hat{j} + \hat{k}, \hat{i} + μ \hat{j} + \hat{k}, \hat{i} + \hat{j} + μ \hat{k}$ will be coplanar, if

$\begin{aligned} | \begin{array}{ccc} μ & 1 & 1 \\ 1 & μ & 1 \\ 1 & 1 & μ \end{array} | = 0 \\ \Rightarrow & μ (μ^{2} - 1) - 1 (μ - 1) + 1 (1 - μ) = 0 \\ \Rightarrow & (μ - 1) [μ (μ + 1) - 1 - 1] = 0 \\ \Rightarrow & (μ - 1) [μ^{2} + μ - 2] = 0 \\ \Rightarrow & (μ - 1) [(μ + 2) (μ - 1)] = 0 \\ \Rightarrow & μ = 1 or - 2 \end{aligned}$

So, sum of the distinct real values of $μ = 1 - 2 = - 1$ .

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Vectors 3 Question 1

1. The sum of the distinct real values of μ, for which the vectors, μi^+j^+k^,i^+μj^+k^,i^+j^+μk^ are coplanar, is

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1. The sum of the distinct real values of $μ$ , for which the vectors, $μ \hat{i} + \hat{j} + \hat{k}, \hat{i} + μ \hat{j} + \hat{k}, \hat{i} + \hat{j} + μ \hat{k}$ are coplanar, is