Question

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

Answer 1

Answer:

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

        $v  =    |v| u =  \sqrt{12} (\frac{2}{ \sqrt{12} } , -\frac{2}{ \sqrt{12} }, - \frac{2}{ \sqrt{12} })$

 

Step-by-step 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

          $|v| =  \sqrt{ 2^2  + (-2)^2 +  (-2)^2 }$

         $|v| =  \sqrt{12}$

The unit vector is mathematically represented as

        $u  =  \frac{v}{|v|}$

        $u  =  \frac{ <2 , -2 , -2>}{\sqrt{12} }$

    $u =  \frac{2}{ \sqrt{12} } , -\frac{2}{ \sqrt{12} }, - \frac{2}{ \sqrt{12} }$

So

    $v  =    |v| u =  \sqrt{12} (\frac{2}{ \sqrt{12} } , -\frac{2}{ \sqrt{12} }, - \frac{2}{ \sqrt{12} })$