How to find a vector that has the same direction

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Homework Help Overview

The problem involves finding a vector that shares the same direction as the vector <-2, 4, 2> but has a specified length of 6. The context is within vector mathematics, specifically focusing on vector direction and magnitude.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the concept of unit vectors and how to derive a vector of a specific length from an original vector. There are attempts to express the new vector in terms of a scaling factor applied to the original vector.

Discussion Status

Some participants have provided insights into the relationship between the original vector's length and the desired length, exploring the concept of scaling. There is acknowledgment of the uniqueness of the scaling factor, though the discussion does not reach a definitive conclusion.

Contextual Notes

Participants note the importance of understanding the relationship between vector length and direction, as well as the implications of scaling factors in vector mathematics. There is an emphasis on the uniqueness of direction in three-dimensional space.

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

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