# 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

Jorge Bayard | 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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## Related Questions

Lupe Tanner, Ph.D. | 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 | 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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