Find unit vectors that satisfy the given conditions: The unit vector that has the same direction as the vector from the point A=(-4,3) to the point B=(1,7)

Expert Answers

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The vector from (-4,3) to (1,7) is <5,4>, because 1-(-4)=5 and 7-3=4

If we multiply both entries of our vector by the same positive number, we will get a vector in the same direction as our original vector.  (If we multiply both entries of our vector by the same negative number, we will get a vector in the opposite direction as our original vector.)

The length of the vector <5,4> is

`sqrt(5^2+4^2)=sqrt(25+16)=sqrt(41)~~6.40312424`

We would like a unit vector (so length 1).  Our vector has length approx 6.4, so we need to multiply each entry by approx: 1/6.4

`<5/sqrt(41),4/sqrt(41)>`

To simplify, multiply the numerators and denominators by `sqrt(41)`

`<(5 sqrt(41))/41, (4 sqrt(41))/41>`

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