Find a unit vector in the direction of the given vector. v=<5,10> ​(Type your answer in the form ai​ + bj. Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

Question

Find a unit vector in the direction of the given vector.

v=<5,10>

​(Type your answer in the form ai​ + bj. Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

Answer 1

A unit vector has magnitude 1, so to get a unit vector from a given vector, you can scale the given vector by 1/(it’s magnitude).

|(5, 10)| = √(5² + 10²) = √125 = 5√5

The unit vector is 1/5√5 (5, 10) = (1/√5, 2/√5)

In the form ai + bj, it’s (1/√5)i + (2/√5)j

(1/√5)i + (2/√5)j

Answer 2

<5,10>

in polar end of the vector from the origin = (5sqr5, 63.435)

magnitude = sqr(5^2+10^2) =sqr125 = 5sqr5

angle = tan^-1(10/5) = 63.435 degrees

to get magnitude 1, the polar coordinates are (1, 63.435)

convert to rectangular coordinates = (cos63.435, sin63.435) = (.477, .894)

unit vector is <.477, .894>

which has magitude 1 and direction 63.435 degrees = .353pi radians

= 1,107 radians