Let a-b-c are units vectors satisfying a-b-c-a-b-c- If a and c are not collinear and a-c-12- then -a-b-c- equals

A×(b×c )=(a×b )×c ⇒ (a·c)b (a. b )c=(a·c)b (b·c )a ⇒(a·b)c=(b·c)a ⇒ a·b=b·c=0 (∵a,c are not collinear) Now, nbsp;| a+b+c |^2= | a |^2+ | b |^2+ |c |^2+2(a·b ...

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Byju's Answer

Standard XII

Mathematics

Vector Triple Product

Let a,b,c are...

Question

Let $\to a, \to b, \to c$ are units vectors satisfying $\to a \times (\to b \times \to c) = (\to a \times \to b) \times \to c .$ If $\to a$ and $\to c$ are not collinear and $\to a \cdot \to c = \frac{1}{2},$ then $| \to a + \to b + \to c |$ equals

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Solution

$\to a \times (\to b \times \to c) = (\to a \times \to b) \times \to c$
$\Rightarrow (\to a \cdot \to c) \to b - (\to a . \to b) \to c = (\to a \cdot \to c) \to b - (\to b \cdot \to c) \to a$
$\Rightarrow (\to a \cdot \to b) \to c = (\to b \cdot \to c) \to a$
$\Rightarrow \to a \cdot \to b = \to b \cdot \to c = 0 (∵ \to a, \to c are not collinear)$

Now, ${∣ ∣ ∣ \to a + \to b + \to c ∣ ∣ ∣}^{2} = {∣ ∣ \to a ∣ ∣}^{2} + {∣ ∣ ∣ \to b ∣ ∣ ∣}^{2} + {∣ ∣ \to c ∣ ∣}^{2} + 2 (\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a = = 3 + 2 (0 + 0 + \frac{1}{2}) (∵ | \to a | = | \to b | = | \to c | = 1)$
$= 4$
$∴ ∣ ∣ ∣ \to a + \to b + \to c ∣ ∣ ∣ = \sqrt{4} = 2$

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Similar questions

Q. If

\to a, \to b, \to c

are unit vectors and

\to b, \to c

are non-collinear vectors satisfying

(\to a, \to b) = α, (\to a, \to c) = β

and

\to a \times (\to b \times \to c) = \frac{\to b + \to c}{2},

then

cos (α + β) =

Q. If

\to b and \to c

are two non-collinear unit vectors and

\to a

is any vector, then

(\to a \cdot \to b) \to b + (\to a \cdot \to c) \to c + \frac{\to a \cdot (\to b \times \to c)}{| \to b \times \to c |^{2}} (\to b \times \to c) =

Q. If

\to b

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is any vector, then

(\to a \cdot \to b) \to b + (\to a \cdot \to c) \to c + ⎡ ⎢ ⎣ \frac{\to a \cdot (\to b \times \to c)}{| \to b \times \to c |} ⎤ ⎥ ⎦ (\to b \times \to c) =

Q. If

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and

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| \to a - \to b |^{2} + | \to b - \to c |^{2} + | \to c - \to a |^{2} = 9,

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Q. If

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and

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\to p, \to q

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\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]}

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Let →a,→b,→c are units vectors satisfying →a×(→b×→c)=(→a×→b)×→c. If →a and →c are not collinear and →a⋅→c=12, then |→a+→b+→c| equals

Let $\to a, \to b, \to c$ are units vectors satisfying $\to a \times (\to b \times \to c) = (\to a \times \to b) \times \to c .$ If $\to a$ and $\to c$ are not collinear and $\to a \cdot \to c = \frac{1}{2},$ then $| \to a + \to b + \to c |$ equals