# 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 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...

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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

Approved by eNotes Editorial Team 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.

Approved by eNotes Editorial Team 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

Approved by eNotes Editorial Team