If a- b- c- are three non-coplanar vectors such that r-1-a-b-c- r-2-b-c-a- r-3-c-a-b- r-2a-3b-4c- if r-1r-1-2r-2-3r-3- then

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Applications of Dot Product

If a⃗, b⃗, ...

Question

If $\to a, \to b, \to c$ are three non-coplanar vectors such that
$_{1} = \to a - \to b + \to c,_{2} = \to b + \to c - \to a$ ,
$_{3} = \to c + \to a + \to b, \to r = 2 \to a - 3 \to b + 4 \to c$ if
$\to r = λ_{1}_{1} + λ_{2}_{2} + λ_{3}_{3}$ , then

λ_{1} = 7 / 2

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λ_{1} + λ_{2} = 3

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λ_{1} + λ_{2} + λ_{3} = 4

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λ_{2} + λ_{3} = 2

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Solution

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\to a, \to b, \to c

_{1} = \to a - \to b + \to c,_{2} = \to b + \to c - \to a,_{3} = \to c + \to a + \to b

\to r = 2 \to a - 3 \to b + 4 \to c

\to r = λ_{1}_{1} + λ_{2}_{2} + λ_{3}_{3}

\to a, \to b, \to c

_{1} = \to a - \to b + \to c,_{2} = \to b + \to c - \to a

_{3} = \to c + \to a - \to b, \to r = 2 \to a - 2 \to b + 4 \to c

\to r = x_{1}_{1} + x_{2}_{2} + x_{3}_{3}

\to a, \to b, \to c

\to r

(\to a \times \to b) \times (\to r \times \to c) + (\to b \times \to c) \times (\to r \times \to a) + (\to c \times \to a) \times (\to r \times \to b) =

\to a, \to b, \to c

\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]}

(2 \to a + 3 \to b + 4 \to c) . \to p + (2 \to b + 3 \to c + 4 \to a) . \to q +

(2 \to c + 3 \to a + 4 \to b) . \to r =

\to a, \to b, \to c

\to r

(\to a \times \to b) \times (\to r \times \to c) + (\to b \times \to c) \times (\to r \times \to a) + (\to c \times \to a) \times (\to r \times \to b)

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