Prove that a necessary and sufficient condition for three vectors a - b - and c - to be coplanar is that there exist scalars l- m- n not all zero simultaneously such that la - mb - nc -0 -

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Condition for Concurrency of Three Lines

Prove that a ...

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Prove that a necessary and sufficient condition for three vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that $l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .$

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\to a \times (\to b \times \to c), \to b \times (\to c \times \to a), \to c \times (\to a \times \to b)

l, m, n

l \to a + m \to b + n \to c = \to 0

\to a, \to b, \to c

\to a, \to b, \to c

\to p, \to q, \to r

\to a, \to b, \to c

(l \to a + m \to b + n \to c) . (l \to p + m \to q + n \to r)

: (

l, m, n

)

(l a + m b + n c) . (l p + m q + n r)

\frac{x}{l (m b + n c - l a)} = \frac{y}{m (n c + l a - m b)} = \frac{z}{n (l a + m b - n c)},

\frac{l}{x (b y + c z - a x)} = \frac{m}{y (c z + a x - b y)} = \frac{n}{z (a x + b y - c z)} .

Prove that the necessary and sufficient condition for an integer n to odd is that $n^{2}$ is odd.

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