# Find the angle alpha in radians between the vector x= <2,2,2,-4>and vector y = <-1,-2,-3,1>

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You need to use the dot product definition to evaluate the angle `alpha` , such that:

`bar x*bar y = |bar x|*|bar y|*cos alpha`

`cos alpha = (bar x*bar y)/(|bar x|*|bar y|)`

You need to perform the multiplication of vectors, such that:

`bar x*bar y = 2*(-1) + 2*(-2) + 2*(-3) + (-4)*1`

`bar x*bar y = -2 - 4 - 6 - 4 = -16`

You need to evaluate the lengths of the vectors `bar x` and `bar y` such that:

`|bar x| = sqrt(2^2 + 2^2 + 2^2 + (-4)^2) = sqrt28 => |bar x| = 2sqrt7`

`|bar y| = sqrt((-1)^2 + (-2)^2 + (-3)^2 + 1^2) => |bar y| = sqrt15`

`cos alpha = (-16)/(2sqrt7*sqrt15) => cos alpha = -8/sqrt105 => cos alpha = -0.780 => alpha ~~ (11pi)/24.`

**Hence, evaluating the angle alpha between the given vectors yields **`alpha ~~ (11pi)/24.`