Let vector A=2x-2y and vector B= 2x+2y. What is the value for the angle that lies between two vectors?
To find the angle between two vectors A and B, we can use the formula:
cosθ = A • B / |A| * |B|
Where A • B is the dot product of vectors A and B, and |A| and |B| are the magnitudes of vectors A and B respectively.
First, let's find the dot product of vectors A and B:
A • B = (2x - 2y) • (2x + 2y)
= 2x * 2x + 2x * 2y - 2y * 2x - 2y * 2y
= 4x^2 + 4xy - 4xy - 4y^2
= 4x^2 - 4y^2
Next, let's find the magnitudes of vectors A and B:
|A| = √((2^2) + (-2)^2) = √(4 + 4) = √8 = 2√2
|B| = √((2^2) + 2^2) = √(4 + 4) = √8 = 2√2
Now, we can plug these values back into the formula for the cosine of the angle between the two vectors:
cosθ = (4x^2 - 4y^2) / (2√2 * 2√2)
= (4x^2 - 4y^2) / 8
= (x^2 - y^2) / 2
To find the angle θ, we need to take the inverse cosine of this value:
θ = cos^(-1)((x^2 - y^2) / 2)
Therefore, the angle that lies between the vectors A=2x-2y and B=2x+2y is θ = cos^(-1)((x^2 - y^2) / 2).