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Multiplication with Vectors

If for three ...

Question

If for three non-zero vectors $\to a, \to b$ and $\to c, \to a \cdot \to b = \to a \cdot \to c$ and $\to a \times \to b = \to a \times \to c$ , then show that $\to b = \to c$

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Solution

Consider the problem
Given.

$\begin{matrix} \to a \cdot \to b = \to a \cdot \to c \to a \cdot \to b - \to a \cdot \to c = 0 \to a \cdot (\to b - \to c) = 0 \end{matrix}$

$\begin{matrix} \to a \times \to b = \to a \times \to c \to a \times \to b - \to a \times \to c = 0 \to a \times (\to b - \to c) = 0 \to a = 0 o r \to b - \to c = 0 \to b - \to c = 0 (\to a \neq 0) \to b = \to c \end{matrix}$

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Similar questions

\to a, \to b, \to c

are three unit vectors such that

\to a \cdot \to b = \to a \cdot \to c = 0

and the angle between

\to b

\to c

\frac{π}{3},

then the value of

| \to a \times \to b - \to a \times \to c |

\to a, \to b, \to c

are non-coplanar non-zero vectors such that

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

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

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

\to a, \to b, \to c

are non-zero non-collinear vectors such that

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

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

Q. For non-zero vectors

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

holds if and only if:

\to a, \to b, \to c

are unit vectors such that

\to a \cdot \to b = 0 = \to a \cdot \to c

and the angle between

\to b

\to c

\frac{π}{3}

, then the value of

| \to a \times \to b - \to a \times \to c |

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