If a- b-c are non-coplanar vectors and a vector V satisfying V-a-V-b-V-c-0- then -V- is equal to

Any vector V can be written in terms of three non coplanar vectors hence, V=p(a×b)+q(b×c)+r(c×a) where (a×b), (b×c),(c×a) are non coplanar vectors and p,q,r are ...

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Standard XII

Mathematics

Vector Triple Product

If a, b,c are...

Question

If $\to a, \to b, \to c$ are non-coplanar vectors and a vector $\to V$ satisfying $\to V \cdot \to a = \to V \cdot \to b = \to V \cdot \to c = 0,$ then $| \to V |$ is equal to

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Solution

Any vector $\to V$ can be written in terms of three non-coplanar vectors
hence,
$\to V = p (\to a \times \to b) + q (\to b \times \to c) + r (\to c \times \to a)$
where $(\to a \times \to b), (\to b \times \to c), (\to c \times \to a)$ are non-coplanar vectors and $p, q, r$ are scalars.
now, taking dot with $\to a :$
we have,
$\to V \cdot \to a = q [\to b \to c \to a] = 0 \Rightarrow q = 0,$
similarly,
$\to V \cdot \to b = r [\to b \to c \to a] = 0 \Rightarrow r = 0$
and
$\to V \cdot \to c = p [\to a \to b \to c] = 0 \Rightarrow p = 0$
So, $\to V = \to 0$
$\Rightarrow | \to V | = 0$

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Vector Triple Product

Standard XII Mathematics

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If →a,→b,→c are non-coplanar vectors and a vector →V satisfying →V⋅→a=→V⋅→b=→V⋅→c=0, then |→V| is equal to

If $\to a, \to b, \to c$ are non-coplanar vectors and a vector $\to V$ satisfying $\to V \cdot \to a = \to V \cdot \to b = \to V \cdot \to c = 0,$ then $| \to V |$ is equal to