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Applications of Cross Product

If a⃗,b⃗,c⃗...

Question

If $\to a, \to b, \to c$ are vectors such that $\to a . \to b = \to a . \to c, \to a \times \to b = \to a \times \to c, \to a \neq 0$ , Then show that $\to b = \to c$ .

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Solution

$\begin{matrix} \to a . \to b = \to a . \to c \to a . \to b = \to a . \to c = 0 \to a (\to b - \to c) = 0 | a | | b - c | cos θ = 0 cos θ \neq 0 | b - c | = 0 b = c \end{matrix}$

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

\to a, \to b

\to c

are any three vectors such that

\to a . \to b \neq 0, \to b . \to c \neq 0

\to a

\to c

\to a, \to b, \to c

are vectors such that

\to a . \to b = 0

\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

are three vectors such that

\to a, \to b, \to c

, then prove that

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

, and hence show that

[\to a \to b \to c] = 0.

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