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Thermal Resistance

If a + b + c ...

Question

If $\to a + \to b + \to c = 0$ , then show that $\to a \times \to b = \to b \times \to c = \to c \times \to a$ . Interpret the result geometrically.

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Solution

$Since, \to a + \to b + \to c = 0 \Rightarrow \to b = - \to c - \to a Now, \to a \times \to b = \to a \times (- \to c - \to a) = \to a \times (- \to c) + \to a \times (- \to a) = - \to a \times \to c \Rightarrow \to a \times \to b = \to c \times \to a . . . (i) A l s o, \to b \times \to c = (- \to c - \to a) \times \to c = (- \to c \times \to c) + (- \to a \times \to c) = - \to a \times \to c \Rightarrow \to b \times \to c = \to c \times \to a . . . (i i) From Eqs.(i) and (ii), \to a \times \to b = \to b \times \to c = \to c \times \to a$

Geometrical interpretation of the result

If ABCD is a parallelogram such that $- - \to A B = \to a a n d - - \to A D = \to b$ and these adjacent sides are making angle $θ$ between each other, then we say that

Area of parallelogram ABCD = $| \to a | | \to b | | s i n θ | = | \to a \times \to b |$

Since, parallelogram on the same base and between the same parallels are equal in area.

We can say that, $| \to a \times \to b | = | \to a \times \to c | = | \to b \times \to c |$

This also implies that, $\to a \times \to b = \to a \times \to c = \to b \times \to c$

So, area of the parallelograms formed by taking any two sides represented by $\to a, \to b a n d \to c$ as adjacent are equal.

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