Quick Vector Question: Finding Unit Vectors and Direction | Explanation Included

[tnutty]

Homework Statement

Find a unit vector that has the same direction as the given vector.

8i - j + 4k

The Attempt at a Solution

I got

sqrt(81) * (8i - j + 4k). The book has 1/9 * (8i-j+4k).

Why exactly is that?


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Another question :

Find a vector that has the same direction as -2i+4j+2z but has length 6.

I think I could find the unit vector by dividing each component by its magnitude, and
then multiply it by a scaler number, 6.

Can you comment on that?

 

[rock.freak667]

any vector parallel or in the same direction as 8i - j + 4k has a vector form of λ(8i - j + 4k) where λ=scalar.


So all vectors parallel to 8i - j + 4k can be written as 8λi-λj+4λk


Unit vector means magnitude=1

so |8λi-λj+4λk|=1

you can find λ from the definition of the magnitude of a vector.

For the next question, you can find it like that as well.

 

[LCKurtz]

Homework Statement

Find a unit vector that has the same direction as the given vector.

8i - j + 4k

The Attempt at a Solution

I got

sqrt(81) * (8i - j + 4k). The book has 1/9 * (8i-j+4k).

Why exactly is that?


Because you forgot to divide the vector by its length instead of multiply, and you have forgotton what sqrt(81) is.