What is the meaning of magnitude of a vector and unit v?

Question

What is the meaning of magnitude of a vector and unit v?

Answer 1

the magnitude of a vector is how long it is

a unit vector is a vector of unit 1

Answer 2

Magnitude is how long the vector is (a scalar quantity).

If you take the square root of the sum of the squares of its components, you get its magnitude.

Two dimensions:

r = √( a² + b² )

Three dimensions:

r = √( a² + b² + c² )

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A unit vector has a magnitude of 1.

If you want to make a unit vector out of another vector, you divide the components by the magnitude (a scalar).

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For example:

A vector, which is often written as one of the following:

< 2, 1, 1 >

= 2·i + j + 2·k

The magnitude would be:

r = √( 2² + 1² + 2² )

= √( 4 + 1 + 4 )

= √( 9 )

= 3

A unit vector would be found by dividing by the magnitude above (3):

(1/3) · ( 2·i + j + 2·k )

= (2/3)·i + (1/3)·j + (2/3)·k