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Class 10 Class 12
Establish the vector inequalities geometrically or otherwise :
(c) |ab| < |a| + |b|

Let two vectors a and b be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

Here we have:

| OR | = | PS | = | b |              ...(i)

| OP | = | a |                          ...(ii)

In a triangle, each side is smaller than the sum of the other two sides. Therefore, in ΔOPS, we have:

OS < OP + PS

| a - b | < | a | + | -b |

| a - b | < | a | + | b |            ... (iii)

If the two vectors act in a straight line but in opposite directions, then we have,

| a - b | = | a | + | b |             ... (iv)

Combining equations (iii) and (iv), we get,

| a - b | ≤ | a | + | b |
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What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors.

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Give three examples of scalar quantities.

Mass, temperature and energy
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Give three examples of vector quantities.

Force, impulse and momentum.
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What is a scalar quantity?

A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
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What is a vector quantity?

A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
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