Calculating angles using vectors?

For the vectors


u =


3


1


and


v =


4


5


calculate (a) The angle between u and the x axis


(b) The angle between v and the x axis


(c) The angle, θ (theta), between u and v


(d) The length of u |u|


(e) The lenght of v |v|


(f) The value of |u| * |v| * cos θ


(g) The value of u*v

Calculating angles using vectors?
Given vectors u and v


u = 3i + j


v = 4i + 5j





Calculate:





(a) The angle α between u and the x axis.


α = arctan(1/3)





(b) The angle β between v and the x axis.


β = arctan(5/4)





(c) The angle θ between u and v.


θ = β - α





tanθ = tan(β - α) = (tanβ - tanα) / (1 + tanα tanβ)


tanθ = (5/4 - 1/3) / [1 - (5/4)(1/3)] = (15/12 - 4/12) / (1 + 5/12)


tanθ = (11/12) / (17/12) = 11/17





θ = arctan(11/17)





(d) The length of u.


| u | = √(3² + 1²) = √(9 + 1) = √10





(e) The lenght of v.


| v | = √(4² + 5²) = √(16 + 25) = √41





(f) The value of |u| * |v| * cos θ





tanθ = sinθ/cosθ = 11/17





sin²θ + cos²θ = 11²/x² + 17²/x² = 1


121 + 289 = x²


x² = 410


x = √410





cosθ = 17/√410





|u| * |v| * cos θ = (√10)(√41)(17/√410) = 17





(g) The value of u•v





u = %26lt;3, 1%26gt;


v = %26lt;4, 5%26gt;





u • v = %26lt;3, 1%26gt; • %26lt;4, 5%26gt; = 12 + 5 = 17
Reply:cos(theta)=u dot v/ magnitude of u times v





magnitude u equals sqrt(x^2 + y^2)





dot product equals product of x components plus product of y components of the two vectors.
Reply:This is clearly intended as a "discovery" exercise.


(a) Simplest approach is to use the inverse tangent. Since the tangent of a standard position angle is the y over the x...


angle = arctan(1/3)


(b) angle = arctan(5/4)


(c) subtract (a) from (b) to find theta.


(d) |u| is the magnitude of u. Basically use the pythagorean theorem to find the length of the vector: sqrt(10).


(e) Same here: sqrt(41)


(f) You can handle this one. :-)


(g) u*v (I'm assuming from context you mean the dot product) is calculated by multiplying the x-components together and adding the product of the y-components: 3 * 4 + 1 * 5





You should get the same answer as you did for (f)!!!





Good luck!
Reply:a) u = (3i + j)


x = i


u.x = |u||x| cos a


where a is the angle between u and x axis


3 = sqrt(10) cos a


=%26gt; a = cos ^-1 3/sqrt(10)





b) v.x = |v||x| cos a


where a is the angle between v and x axis


a = cos ^-1 4/sqrt(41)





c) |u| = sqrt(10)





d) |v| = sqrt (41)





e) |u| * |v| * cos θ = u.v = 12 + 5 = 17





f) u*v =


| i j k |


| 3 1 0 |


| 4 5 0 |





= 11 perpendicular to both i and j