Prove that- a- b- - c- a- - 2b- - 3c- - -a-b-c-

L.H.S. = a⃗· (b⃗ + c⃗)× (a⃗ + 2b⃗ + 3c⃗) = a⃗· (b⃗× #160;b⃗× 2b⃗ + b⃗× 3c⃗ + c⃗×a⃗ + c⃗× 2b⃗ + c⃗× 3c⃗) = a⃗· (b⃗×a⃗ + 3b⃗×c⃗ + c⃗×a⃗ + 2c⃗×b⃗) ...

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Standard XII

Mathematics

Definition of Vector

Prove that: ...

Question

Prove that: $\to a \cdot (\to b + \to c) \times (\to a + 2 \to b + 3 \to c) = [\to a \to b \to c]$

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Solution

$L . H . S . = \to a \cdot (\to b + \to c) \times (\to a + 2 \to b + 3 \to c)$
$= \to a \cdot (\to b \times \to b \times 2 \to b + \to b \times 3 \to c + \to c \times \to a + \to c \times 2 \to b + \to c \times 3 \to c)$
$= \to a \cdot (\to b \times \to a + 3 \to b \times \to c + \to c \times \to a + 2 \to c \times \to b)$
$= \to a \cdot (\to b \times \to a) + \to a \cdot (3 \to b \times \to c) + \to a \cdot (\to c \times \to a) + \to c \cdot (2 \to c \times \to b)$
$= 0 + 3 [\to a \cdot (\to b \times \to c) + 0 + 2 (\to a \cdot (\to c \times \to b)]$
$= 3 [\to a \to b \to c] + 2 [\to a \to b \to c]$
$= 3 [\to a \to b \to c] - 2 [\to a \to b \to c]$
$= [\to a \to b \to c] = R . H . S .$

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Prove that: →a⋅(→b+→c)×(→a+2→b+3→c)=[→a→b→c]

Prove that: $\to a \cdot (\to b + \to c) \times (\to a + 2 \to b + 3 \to c) = [\to a \to b \to c]$