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Applications of Cross Product

If αa⃗×b⃗+β...

Question

If $α (\to a \times \to b) + β (\to b \times \to c) + γ (\to c \times \to a) = \to 0$ and at least one of the scalars $α, β, γ$ is non-zero, then the vectors $\to a, \to b, \to c$ are

A

Collinear

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B

Coplanar

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C

Non-coplanar

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D

Cannot be determined.

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Solution

The correct option is C Coplanar
As per the question, and the definition of linearly dependent vectors, we get that $(¯ ¯ ¯ a \times ¯ ¯ b), (¯ ¯ ¯ a \times ¯ ¯ c)$ and $(¯ ¯ b \times ¯ ¯ c)$ are linearly dependent.
Now, since they are linearly dependent, any one of them can be written in terms of other two and thus we get that they are coplanar.

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Similar questions

α \to a + β \to b + γ \to c = \to 0

(\to a \times \to b) \times {(\to b \times \to c) \times (\to c \times \to a} =

\to a = ˆ i + 2^j + 3^k, \to b = 2^i + 3^j +^k, \to c = 3^i +^j + 2^k

are vectors satisfying

α \to a + β \to b + γ \to c = - 3 (^i -^k)

, then the ordered triplet

(α, β, γ)

\to a =^1 + 2^j + 3^k, \to b = 2^i + 3^j +^k, \to c = 3^i +^j + 2^k

α \to a + β \to b + γ \to c = - 3 (^i -^k) \cdot

Then the triplet

(α, β, γ)

\to a =^i +^j, \to b =^j +^k, \to c = α \to a + β \to b

and the vectors

^i - 2^j +^k, 3^i + 2^j -^k, \to c

are coplanar, then

\frac{α}{β} =

Q. Statement-I :

\to a \times (\to b \times \to c), \to b \times (\to c \times \to a), \to c \times (\to a \times \to b)

are coplanar vectors.

Statement-II: If there exists scalars

l, m, n

not all zero such that

l \to a + m \to b + n \to c = \to 0

, then the vectors

\to a, \to b, \to c

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