Why is a 12 N force along 2i+j-2k written as 8i+4j-8k?

[physics kiddy]

Our teacher taught us a way to represent a force as a unit vector.

Suppose a force of 12 N acts along the line 2i+j-2k. The force is written as:

F = 12 N (2i+j-2k/√1^2+2^2+2^2)

Therefore,

F = 8i+4j-8k

But, I can't understand why is it so. Please explain.

 

[tiny-tim]

hi physics kiddy!

let's take an easier 2D case, a force of 15 N acts along the line 3i + 4j

if you draw 3i + 4j, you can see that it has magnitude 5

so 3i + 4j represents a force of magnitude 5 N along that line

to get a force of magnitude 15 N, we need 3 times that,

= 3 N (3i + 4j)

= 15 N (3i + 4j)/√(3² + 4²)​

the magnitude of the vector must equal the magnitude of the force, so we must always multiply the force by a unit vector …

(ai + bj +ck)/√(a² + b² + c²) is always a unit vector ​