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Find a unit vector in the direction of the given vector.  Verify that the result has a magnitude of 1. u = <0, -2> v = <5, -12> w = 7j - 3i

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Jorge Bayard eNotes educator | Certified Educator

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The unit vector of a given vector is defined as the ratio between the vector and its absolute value.

r̂ = r̄/|r̄|

where:

r̂  →  Unit vector in the direction of  r̄

|r̄|  →  Absolute value of  r̄

For the vector ū, we have:

û = ū/|ū|

û = (0 – 2)/√(0^2 + (-2)^2) = (0 – 2)/2

û = <0 - 1>

Verifying that the result has a magnitude of 1:

|û|= √(0^2 + (-1)^2) = 1

For the vector v̄:

v̂ = v̄/|v̄|

v̂ = (5 – 12)/√(5^2 + (-12)^2) = (5 – 12)/13

v̂ = <(5/13) - (12/13)>

|v̂|= √(5/13)^2 + (-12/13)^2) = √(25/169) + (144/169) = 1

For the vector w̄:

w̄ = w̄/|w|

ŵ = -3î + 7ĵ/√(-3^2 + 7^2) = (-3î + 7ĵ)/√58

ŵ = (-3/√58)^2)î - (7/√58)^2)ĵ

|ŵ|= √(9/58) + (49/58) = √(58/58) = 1

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Lupe Tanner, Ph.D. eNotes educator | Certified Educator

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The unit vector is found by diving the given vector with its magnitude (which is the square root of sum of the squares of all the vector coordinates).

Given, u = <0, -2>

Magnitude of u = `sqrt(0^(2) + (-2)^(2))` = `sqrt(4)` = 2

Thus, the unit vector in direction of u will be: <0/2, -2/2> = <0, -1>

and the magnitude of the unit vector =`sqrt(0^(2) + (-1)^(2))` = `sqrt(1)` = 1 

Similarly, for v = <5, -12>

Magnitude of v = `sqrt(5^(2) + (-12)^(2))` = `sqrt(25+144)` = `sqrt(169)` = 13

Unit vector in direction of v = <5/13, -12/13>

 Its magnitude = `sqrt((5/13)^(2) + (-12/13)^(2))`  = `sqrt(25/169 + 144/169)` =  `sqrt (169/169)` = 1  

And for w = 7j -3i

Magnitude = `sqrt (7^(2) + (-3)^(2))` = `sqrt(49+9)`  = `sqrt(58)`

 Thus the unit vector in direction of w will be:

 `7/sqrt(58) j - 3/sqrt(58) i`

Magnitude = `sqrt ((7/sqrt(58))^(2) + (3/sqrt(58))^(2))`  = `sqrt (49/58 + 9/58)` = `sqrt (58/58)`  = 1

Hope this helps.

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nick-teal eNotes educator | Certified Educator

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In order to find a unit vector we divide each component by the magnitude of the vector.

u = <0, -2> magnitude = `sqrt(0^2 + (-2)^2) = 2`

unit vector for u is <0, -1>

v = <5, -12> magnitude = `sqrt(5^2 + (-12)^2) = 13`

unit vector for v is <5/13, -12/13>

w = 7j - 3i magnitude = `sqrt(7^2 + (-3)^2) = 7.615`

unit vector for w is 7/7.615j - 3/7.615i

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