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Fundamental Theorem In 3D

If a ,b , a...

Question

If $\to a, \to b,$ and $\to c$ are unit coplanar vectors then find the value of $[2 \to a - \to b 2 \to b - \to c 2 \to c - \to a] .$

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Solution

$\to a, \to b, \to c$ are coplanar vectors
So [a, b, c] = 0 $\Rightarrow ˆ a \cdot (ˆ b \times ˆ c) = ˆ b (ˆ c \times ˆ a) = ˆ c . (ˆ a \times ˆ b) = 0$
Now
[2a – b, 2b – a , 2c – a]
= $(2a−b)⋅{(2a−b)⋅(2^b−^c)×(2^c−^a)}$
$= (2 ˆ a - ˆ b) [2 ˆ b \times (2 c - a) - ˆ c (2 c - a)]$
$= (2 ˆ a - ˆ b) . [4 ˆ b \times ˆ c - 2 ˆ b \times ˆ a - ˆ c \times 2 ˆ c + ˆ c \times ˆ a)$
$= (2 ˆ a - ˆ b) . [4 ˆ b \times ˆ c + 2 ˆ a \times ˆ b + ˆ c \times ˆ a)$
$= 8 ˆ a (ˆ b \times ˆ c) - ˆ b \cdot (4 b \times c) + 2 ˆ a \cdot (ˆ a \times ˆ b) - 2 ˆ b \cdot (ˆ a \times ˆ b) + 2 ˆ a \cdot (ˆ c \times ˆ a) - ˆ b (ˆ c \times ˆ a)$
$= 8 ˆ a (ˆ b \times ˆ c) - ˆ b \cdot (4 b \times c) + 2 ˆ a \cdot (ˆ a \times ˆ b) - 2 ˆ b \cdot (ˆ a \times ˆ b) + 2 ˆ a \cdot (ˆ c \times ˆ a) - ˆ b (ˆ c \times ˆ a)$
$= 2 ˆ a \cdot (ˆ b \times ˆ c) - ˆ a (ˆ b \times ˆ c)$
$= 7 ˆ a . (ˆ b \times ˆ c)$
$= 0$

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\to a, \to b

\to c

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\frac{[\to a + 2 \to b \to b + 2 \to c \to c + 2 \to a]}{[\to a \to b \to c]} =

\to a, \to b, \to c

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∣ ∣ ∣ (\to a .^d) (\to b \times \to c) + (\to b .^d) (\to c \times \to a) + (\to c .^d) (\to a \times \to b) ∣ ∣ ∣

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{∣ ∣ ∣ \to a - \to b ∣ ∣ ∣}^{2} + {∣ ∣ ∣ \to b - \to c ∣ ∣ ∣}^{2} + {∣ ∣ \to c - \to a ∣ ∣}^{2}

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