Units and Measurement

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Question

Show that a. (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c.

Solution

A parallelepiped with origin O and sides a, b, and c is shown in the following figure.

Volume space of space the space given space parallelopiped space equals space abc space OC with rightwards harpoon with barb upwards on top space space equals space straight a with rightwards harpoon with barb upwards on top space stack OB space with rightwards harpoon with barb upwards on top space equals space straight b with rightwards harpoon with barb upwards on top OC with rightwards harpoon with barb upwards on top space space equals space straight c with rightwards harpoon with barb upwards on top Let space straight n with hat on top space be space straight a space unit space vector space perpendicular space to space both space straight b space and space straight c. space Hence comma space straight n with hat on top space and space straight a space have space the space same space direction. space therefore space straight b with rightwards harpoon with barb upwards on top space straight x space straight c with rightwards harpoon with barb upwards on top space equals space bc space Sin space straight theta space straight n with hat on top space space space space space space space space space space space space space space space space equals space bc space straight n with hat on top space space space space space space space space space space space space space space space space space equals space straight a with rightwards harpoon with barb upwards on top. space open parentheses straight b with rightwards harpoon with barb upwards on top space straight x space straight c with rightwards harpoon with barb upwards on top space close parentheses space space space space space space space space space space space space space space space space equals straight a. space left parenthesis bc space straight n with hat on top space right parenthesis space space space space space space space space space space space space space space space space equals space abc space Cosθ space straight n with hat on top space space space space space space space space space space space space space space space space equals space abc space cos space 0 space space space space space space space space space space space space space space space space space equals space abc space space space space space space space space space space space space space space space space equals space Volume space of space the space parallelopiped space

Hence proved.

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Units And Measurement

Show that a. (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c.

Some More Questions From Units and Measurement Chapter

Define physical quantity.

Give four examples of physical quantities.

Define unit.

What are the two main required characteristics of a unit?

Can we use same units for different physical quantities?

Can we use different units to measure same physical quantity?

To measure a physical quantity X, the unit U is repeatedly used for n times. What is the magnitude of physical quantity?

Does the numerical factor in the quantitative measurement depend on the system of unit used?

Is one unit sufficient to measure a physical quantity?

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