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Applications of Cross Product

If a⃗, b⃗, ...

Question

If $\to a, \to b, \to c$ be the vectors represented by the sides of a triangle taken in order, then prove that $\to a + \to b + \to c = 0$

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Solution

Let $\to A B = \to a, \to B C = \to b, \to C A = \to c$

Then, by triangle law of vector addition

$\to A B + \to B C = \to A C$

$\to A B + \to B C - \to A C = 0$

$\to A B + \to B C + \to C A = 0$ $(\to A B = - \to B A)$

$\to a + \to b + \to c = 0$

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Similar questions

\to a, \to b, \to c

are three vectors such that

\to a \times \to b = \to c, \to b \times \to c = \to a, \to c \times \to a = \to b

then prove that

| \to a | = | \to b | = | \to c |

\to A

\to B

\to C

are vectors such that

| \to B | =

\to C |

[(\to A + \to B) \times (\to A + \to C)] \times (\to B \times \to C) . (\to B + \to C) = 0

\to a + \to b + \to c = 0

, the prove that:

\to a \times \to b = \to b \times \to c = \to c \times \to a

\to a, \to b, \to c

are non-zero vectors.

\to a, \to b, \to c

are three vectors such that

\to a + \to b + \to c = \to 0,

then prove that

\to a \times \to b = \to b \times \to c = \to c \times \to a,

and hence show that

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\to a, \to b, \to c

are three vectors such that

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, then prove that

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, and hence show that

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