Question

How to express a vector as a product of its length and direction?

Answer 1

So the way to express a vector(
`v =  2i - 2j - 2k`) as a product of its length and direction is

`v  =    |v| u =  √(12) ((2)/( √(12) ) , -(2)/( √(12) ), - (2)/( √(12) ))`

Explanation:

Generally a vector is expressed as a product of its length and direction using the formula below

`v  =  |v|\cdot u`

Here v is the vector

|v| is its magnitude (length)

u is its unit vector (direction)

Now let take an example

Let

`v =  2i - 2j - 2k`

The magnitude is mathematically evaluated as

`|v| =  √( 2^2  + (-2)^2 +  (-2)^2 )`

`|v| =  √(12)`

The unit vector is mathematically represented as

`u  =  (v)/(|v|)`

`u  =  ( <2 , -2 , -2>)/(√(12) )`

`u =  (2)/( √(12) ) , -(2)/( √(12) ), - (2)/( √(12) )`

So

`v  =    |v| u =  √(12) ((2)/( √(12) ) , -(2)/( √(12) ), - (2)/( √(12) ))`