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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gsenviro eNotes educator| Certified Educator

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.

electreto05 eNotes educator| Certified Educator

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

nick-teal eNotes educator| Certified Educator

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