# Find the Angle Between the Vectors (1,2.22) , (2,3.32)

(1,2.22) , (2,3.32)
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.
1⋅2+2.22⋅3.32
Simplify.
Remove parentheses.
1⋅2+2.22⋅3.32
Simplify each term.
Multiply 2 by 1.
2+2.22⋅3.32
Multiply 2.22 by 3.32.
2+7.3704
2+7.3704
9.3704
9.3704
9.3704
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.
(1)2+(2.22)2
Simplify.
One to any power is one.
1+(2.22)2
Raise 2.22 to the power of 2.
1+4.9284
5.9284
5.9284
5.9284
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.
(2)2+(3.32)2
Simplify.
Raise 2 to the power of 2.
4+(3.32)2
Raise 3.32 to the power of 2.
4+11.0224
15.0224
15.0224
15.0224
Substitute the values into the equation for the angle between the vectors.
θ=arccos(9.3704(5.9284)⋅(15.0224))
Simplify.
Simplify the denominator.
Combine using the product rule for radicals.
arccos(9.37045.9284⋅15.0224)
Multiply 5.9284 by 15.0224.
arccos(9.370489.05879616)
arccos(9.370489.05879616)
Multiply 9.370489.05879616 by 89.0587961689.05879616.
arccos(9.370489.05879616⋅89.0587961689.05879616)
Combine and simplify the denominator.
Multiply 9.370489.05879616 and 89.0587961689.05879616.
arccos(9.370489.0587961689.0587961689.05879616)
Raise 89.05879616 to the power of 1.
arccos(9.370489.0587961689.05879616189.05879616)
Raise 89.05879616 to the power of 1.
arccos(9.370489.0587961689.05879616189.058796161)
Use the power rule aman=am+n to combine exponents.
arccos(9.370489.0587961689.058796161+1)
arccos(9.370489.0587961689.058796162)
Rewrite 89.058796162 as 89.05879616.
Use axn=axn to rewrite 89.05879616 as 89.0587961612.
arccos(9.370489.05879616(89.0587961612)2)
Apply the power rule and multiply exponents, (am)n=amn.
arccos(9.370489.0587961689.0587961612⋅2)
Combine 12 and 2.
arccos(9.370489.0587961689.0587961622)
Cancel the common factor of 2.
Cancel the common factor.
arccos(9.370489.0587961689.0587961622)
Divide 1 by 1.
arccos(9.370489.0587961689.058796161)
arccos(9.370489.0587961689.058796161)
Evaluate the exponent.
arccos(9.370489.0587961689.05879616)
arccos(9.370489.0587961689.05879616)
arccos(9.370489.0587961689.05879616)
Multiply 9.3704 by 89.05879616.
arccos(88.4293719289.05879616)
Divide 88.42937192 by 89.05879616.
arccos(0.99293248)
Evaluate arccos(0.99293248).
0.11896095
0.11896095
Find the Angle Between the Vectors (1,2.22) , (2,3.32)