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Linear Combination of Vectors

If a,b are tw...

Question

If $\to a, \to b$ are two vectors such that $| \to a \times \to b | = 2,$ then the value of $[\begin{matrix} \to a & \to b & \to a \times \to b \end{matrix}]$ is

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Solution

We know
$[\begin{matrix} \to P & \to Q & \to R \end{matrix}] = [\begin{matrix} \to Q & \to R & \to P \end{matrix}] = [\begin{matrix} \to R & \to P & \to Q \end{matrix}]$
Therefore,
$[\begin{matrix} \to a & \to b & \to a \times \to b \end{matrix}] = (\to a \times \to b) \cdot (\to a \times \to b) = | \to a \times \to b |^{2} = 4$

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Similar questions

\to a

\to b

are two non zero vectors such that

\to a \times \to b = \to b \times \to a,

Q. For any three vectors

\to a, \to b

\to c

, prove that vectors

\to a - \to b, \to b - \to c

\to c - \to a

\to a, \to b

\to c

are three non-coplanar vectors and

\to p, \to q

\to r

are vectors defined by

\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]}

, then the value of

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

\to a, \to b, \to c

are non - coplanar vectors, then

\frac{a . (b \times c)}{(c \times a) . b} + \frac{b . (a \times c)}{c . (a \times b)} = ?

\to a, \to b, \to c, \to d

are non-zero vectors, then

[\to a \times \to b \to a \times \to c \to d]

can be simplified as:

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Linear Combination of Vectors

Standard XII Mathematics

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