How Do You Find a Vector with a Specific Magnitude in the Opposite Direction?

[mmont012]

Homework Statement

Find a vector in the direction opposite to <-4,1,2>, that has a magnitude of 3.

Homework Equations

I think that I did the first part of the problem correctly:
<-4,1,2>
magnitude= sqrt[ (-4)^2+1^2+2^2 ]
= sqrt(16+1+4)
= sqrt(21)
(-4/sqrt(21), 1/sqrt(21), 2/sqrt(21) )
to get the opposite direction, I would just change the signs
(4/sqrt(21), -1/sqrt(21), -2/sqrt(21) )

But I am confused at where to go from here. What does it mean "that has a magnitude of 3"? I may have approached this problem incorrectly. Any help would be great. Thank you for stopping by.

 

[STEMucator]

You have found a normal vector. The magnitude of a normal vector is $1$.

Perhaps the answer is more trivial than it first appears, just think:

$$\sqrt{a^2 + b^2 + c^2} = 3$$

Select a vector $<a, b, c>$ such that the above is satisfied. Multiplying the original vector by $-1$ will help. In fact, one is the loneliest one, and the only one you'll ever need for this problem.

 

[Chestermiller]

As Zondrina has pointed out, you have correctly determined a unit vector in the desired direction. Now, what do you need to multiply this unit vector by to produce a vector having the same direction, but with a magnitude of 3?

Chet

 

[mmont012]

Ah, okay. I think that I was making this problem more difficult than it actual is. Thank you so much for the help!

 

[Mark44]

Removed "Calc III" from thread title -- this is really a precalc type of problem. Also moved thread to the precalc forum section.

 

[mmont012]

How is this a pre-calc problem if I'm in calc 3 and this is the first time that I've dealt with this? Did my university just not cover this? Or am I behind?

 

[Mark44]

How is this a pre-calc problem if I'm in calc 3 and this is the first time that I've dealt with this? Did my university just not cover this? Or am I behind?

Vectors in R² and R³ are often covered in precalculus with trig courses.

 

[SteamKing]

How is this a pre-calc problem if I'm in calc 3 and this is the first time that I've dealt with this? Did my university just not cover this? Or am I behind?

No limits, no derivatives, no integrals = no calculus.

Your calc 3 course may simply be providing a refresher on basic vector math before jumping into vector calculus.