# Find the Angle Between the Vectors (10,5) , (23,-4)

(10,5) , (23,-4)
The equation for finding the angle between two vectors θ states that the dot product of the two vectors equals the product of the magnitudes of the vectors and the cosine of the angle between them.
u⋅v=|u||v|cos(θ)
Solve the equation for θ.
θ=arc⋅cos(u⋅v|u||v|)
Find the dot product of the vectors.
To find the dot product, find the sum of the products of corresponding components of the vectors.
u⋅v=u1v1+u2v2
Substitute the components of the vectors into the expression.
10⋅23+5⋅-4
Simplify.
Remove parentheses.
10⋅23+5⋅-4
Simplify each term.
Multiply 10 by 23.
230+5⋅-4
Multiply 5 by -4.
230-20
230-20
Subtract 20 from 230.
210
210
210
Find the magnitude of u.
To find the magnitude of the vector, find the square root of the sum of the components of the vector squared.
(u1)2+(u2)2
Substitute the components of the vector into the expression.
(10)2+(5)2
Simplify.
Raise 10 to the power of 2.
100+(5)2
Raise 5 to the power of 2.
100+25
125
Rewrite 125 as 52⋅5.
Factor 25 out of 125.
25(5)
Rewrite 25 as 52.
52⋅5
52⋅5
Pull terms out from under the radical.
55
55
55
Find the magnitude of v.
To find the magnitude of the vector, find the square root of the sum of the components of the vector squared.
(u1)2+(u2)2
Substitute the components of the vector into the expression.
(23)2+(-4)2
Simplify.
Raise 23 to the power of 2.
529+(-4)2
Raise -4 to the power of 2.
529+16
545
545
545
Substitute the values into the equation for the angle between the vectors.
θ=arccos(210(55)⋅(545))
Simplify.
Cancel the common factor of 210 and 5.
Factor 5 out of 210.
arccos(5⋅4255⋅545)
Cancel the common factors.
Factor 5 out of 55⋅545.
arccos(5⋅425(5⋅545))
Cancel the common factor.
arccos(5⋅425(5⋅545))
Rewrite the expression.
arccos(425⋅545)
arccos(425⋅545)
arccos(425⋅545)
Simplify the denominator.
Combine using the product rule for radicals.
arccos(425⋅545)
Multiply 5 by 545.
arccos(422725)
arccos(422725)
Simplify the denominator.
Rewrite 2725 as 52⋅109.
Factor 25 out of 2725.
arccos(4225(109))
Rewrite 25 as 52.
arccos(4252⋅109)
arccos(4252⋅109)
Pull terms out from under the radical.
arccos(425109)
arccos(425109)
Multiply 425109 by 109109.
arccos(425109⋅109109)
Combine and simplify the denominator.
Multiply 425109 and 109109.
arccos(421095109109)
Move 109.
arccos(421095(109109))
Raise 109 to the power of 1.
arccos(421095(1091109))
Raise 109 to the power of 1.
arccos(421095(10911091))
Use the power rule aman=am+n to combine exponents.
arccos(4210951091+1)
arccos(4210951092)
Rewrite 1092 as 109.
Use axn=axn to rewrite 109 as 10912.
arccos(421095(10912)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(421095⋅10912⋅2)
Combine 12 and 2.
arccos(421095⋅10922)
Cancel the common factor of 2.
Cancel the common factor.
arccos(421095⋅10922)
Divide 1 by 1.
arccos(421095⋅1091)
arccos(421095⋅1091)
Evaluate the exponent.
arccos(421095⋅109)
arccos(421095⋅109)
arccos(421095⋅109)
Multiply 5 by 109.
arccos(42109545)
Evaluate arccos(42109545).
0.63583842
0.63583842
Find the Angle Between the Vectors (10,5) , (23,-4)