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

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You need to evaluate the dot prodct `bar x*bar y` such that:

`bar x*bar y = |bar x|*|bar y|*cos(hat(bar x,bar y))`

Performing the multiplication of vectors `bar x` and `bar y` yields:

`bar x*bar y = (2 bar i + 2 bar j + 2 bar k)(- bar i - 2 bar j - 3 bar k)`

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

`bar x*bar y = -2 - 4 - 6 = -12`

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) = sqrt 12`

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

`|bar y| = sqrt 14`

`cos(hat(bar x,bar y)) = (bar x*bar y)/(|bar x|*|bar y|)`

`cos(hat(bar x,bar y)) = -12/(sqrt 12*sqrt 14)`

Reducing duplicate terms yields:

`cos(hat(bar x,bar y)) = -2sqrt42/14 => cos(hat(bar x,bar y)) = -sqrt42/7 = -0.925 => alpha = (hat(bar x,bar y)) ~~ (8pi/7)`

**Hence, evaluating the angle `alpha` in radians, yields **`alpha ~~ (8pi/7).`