The vectors a- - 2a - -3ak- 2a - 1- - 2a - 3- - a - 1k- and 3a - 5 - a - 5- - a - 2 k- are non-coplanar for the range of a in

The correct options are B (0,∞) D (1,∞) a #160; amp; 2a #160; amp; 3a 2a+1 amp; 2a+3 amp; a+1 3a+5 amp; a+5 ...

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Evaluation of a Determinant

The vectors ...

Question

The vectors $a^i + 2 a^j - 3 a^k, (2 a + 1)^i + (2 a + 3)^j + (a + 1)^k$ and $(3 a + 5)^i + (a + 5)^j + (a + 2)^k$ are non-coplanar for the range of $a$ in

{0}

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(0, \infty)

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(- \infty, 1)

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(1, \infty)

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Solution

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a

\to α = a^i + 2 a^j - 3 a^k, \to p = (2 a + 1)^i

+ (2 a + 3)^j + (a + 1)^k

\to r = (3 a + 5)^i + (a + 5)^j + (a + 2)^k

\to p = (a + 1)^i + a^j + a^k, \to q = a^i + (a + 1)^j + a^k

\to r = a^i + a^j + (a + 1)^k, (a \in R)

3 (\to p . \to q)^{2} - λ | \to r \times \to q |^{2} = 0,

λ

\to p = (a + 1)^i + a^j + a^k, \to q = a^i + (a + 1)^j + a^k

\to r = a^i + a^j + (a + 1)^k, (a \in R)

3 (\to p . \to q)^{2} - λ | \to r \times \to q |^{2} = 0,

λ

¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c

3 ¯ ¯ ¯ a - 2 ¯ ¯ b - 4 ¯ ¯ c

- ¯ ¯ ¯ a + 2 ¯ ¯ c

- 2 ¯ ¯ ¯ a + ¯ ¯ b + 3 ¯ ¯ c

_{1} = (3^i + 5^j) m

_{2} = (- 5^i + 3^j) m

_{1} = (4^i - 4^j) {ms}^{- 1}

_{2} = (a^i - 3^j) {ms}^{- 1}

2 s

a

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