Help! how would i find 2 unit vectors that make an angle of 60 degrees with v= <3,4>?

Answer 1

The angle θ, of the given vector v = <3, 4> is given by:

tanθ = 4/3
θ = arctan(4/3) ≈ 53.130102°

Calculate vector w.

tan(θ + 60°) = tan[arctan(4/3) + 60°]
= (4/3 + tan60°) / [1 - (4/3)(tan60°)]
= (4/3 + √3) / [1 - (4/3)(√3)]
= (4 + 3√3) / (3 - 4√3)

θ + 60° = arctan[(4 + 3√3) / (3 - 4√3)] ≈ 113.130102°

w = <3 - 4√3, 4 + 3√3>

|| w || = √[(3 - 4√3)² + (4 + 3√3)²] = √100 = 10

The unit vector is:

w / || w || = <3 - 4√3, 4 + 3√3> / 10
___________

The other vector w is calculated similarly.

Calculate vector w.

tan(θ - 60°) = tan[arctan(4/3) - 60°]
= (4/3 - tan60°) / [1 + (4/3)(tan60°)]
= (4/3 - √3) / [1 + (4/3)(√3)]
= (4 - 3√3) / (3 + 4√3)

θ - 60° = arctan[(4 - 3√3) / (3 + 4√3)] ≈ -6.8698976°

w = <3 + 4√3, 4 - 3√3>

|| w || = √[(3 + 4√3)² + (4 - 3√3)²] = √100 = 10

The unit vector is:

w / || w || = <3 + 4√3, 4 - 3√3> / 10

Answer 2

permit be the style of vector, then a^2 + b^2 = a million because of fact it fairly is a unit vector. additionally, with the aid of the three-4-5 precise triangle <3/5,4/5> is a unit vector in the path of v. So dot <3/5,4/5> = (3/5)a+(4/5)b = cos60 = a million/2. So we've 2 equations in 2 unknowns a,b: a^2 + b^2 = a million (3/5)a+(4/5)b = a million/2 resolve the 2d equation for b and plug into the 1st and get a quadratic equation in a. resolve it getting 2 ideas for a, and plug into the 2d equation getting the two corresponding values of b. this provides: = < (3-4sqrt(3))/10, (4+3sqrt(3))/10 > and = < (3+4sqrt(3))/10, (4-3sqrt(3))/10 >.