How to find a vector that has the same direction

[baokhuyen]

Homework Statement

Can anyone help me with his problem:
how to find a vector that has the same direction as <-2,4,2> but has length 6

Homework Equations

The Attempt at a Solution

the length of that vector is: 6= (-2)^2*a^2+(4)^2*b^2+(2)^2*c^2
Then I don't know what to do next?

 

[ZioX]

Suppose I have a unit vector (a,b,c). What is the length of k(a,b,c)?

 

[Antineutron]

divide the lengh from the original vector, that will give you the unit vector. What do you do with the unit vector to get a lengh of 6 and same direction?

 

[baokhuyen]

k(a,b,c)=(ka,kb,kc)=(-2k,4k,2k)
lenght= root(4k^2+16k^2+4k^2)=6
=> root( 24k^2) =6
=> k*root(24)=6
=>k= 6/root(24)= 3/root6
oh I got it, Thank you very much! Ziox AND Antineutron

 

[ZioX]

Although you haven't done what we said, that works. Note how this scaling quantity is unique up to +/-. Which makes sense: the direction induced by (-2,4,2) is just a line in R^3.

The scaling value that you actually calculated is the ratio of new length to the original length - dividing by the old and multiplying by the new.