A - b - c are non-coplanar vectors such that P - a - b - c - Q -4 a -3 b -4 c and R - a - b - c are linearly dependentvectors- then number of possible values of - is -A- 0B- 1C- 2D- Infinite

The correct option is D InfiniteFor linearly dependent vectors [P Q R]=0 ⇒ 1 amp; 1 amp; 1 nbsp; 4 amp; 3 amp; 4 nbsp; 1 amp; α amp; β ...

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Test for Coplanarity

a , b , c are...

Question

$\to a, \to b, \to c$ are non-coplanar vectors such that $\to P = \to a + \to b + \to c$ , $\to Q = - \to 4 a + \to 3 b + \to 4 c$ and $\to R = \to a + α \to b + β \to c$ are linearly dependent vectors, then number of possible values of $α$ is -

0

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1

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2

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Infinite

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\to a, \to b, \to c

\to p, \to q, \to r

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}, \to r = \frac{\to a \times \to b}{[\to a \to b \to c]},

(\to a + \to b + \to c), (\to p + \to q + \to r)

\to a, \to b, \to c

\to p = \to a + \to b - \to c, \to q = \to b + \to c - \to a, \to r = \to c + \to a - \to b, \to d = 2 \to a - 3 \to b + 4 \to c

\to d = α \to p + β \to q + γ \to r

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}

\to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

\to a, \to b

\to c

(\to a + \to b + \to c) \cdot (\to p + \to q + \to r)

¯ ¯ ¯ a, ¯ ¯ b, ¯ ¯ c

- ¯ ¯ ¯ a + 3 ¯ ¯ b - 5 ¯ ¯ c, - ¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c

2 ¯ ¯ ¯ a - 3 ¯ ¯ b + ¯ ¯ c

\to a, \to b, \to c

\to p, \to q, \to r

\to a . \to p + \to b . \to q + \to c . \to r = 3

\to p

\to q

\to r

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