# Suppose vector u= <2,-1,-4>. Then determine what type of angle the vector u forms with the following vector <3,2,-5>

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You need to use the dot product definition to evaluate the type of angle between the vectors `bar u` and `bar v` , such that:

`bar u*bar v = |bar u|*|bar v|*cos(hat(bar u, bar v))`

You need to identify the vectors `bar u` and `bar v` , such that:

`bar u = 2 bar i - bar j - 4 bar k`

`bar v = 3 bar i + 2 bar j - 5 bar k`

You need to perform the myultiplication of the vectors `bar u` and `bar v` such that:

`bar u * bar v = (2 bar i - bar j - 4 bar k)(3 bar i + 2 bar j - 5 bar k)`

`bar u * bar v = 2*3 - 1*2 + (-4)*(-5)`

`bar u * bar v = 6 - 2 + 20 = 24`

You need to evaluate the lengths of the vectors `bar u` and `bar v` , such that:

`|bar u| = sqrt(2^2 + (-1)^2 + (-4)^2)`

`|bar u| = sqrt 21`

`|bar v| = sqrt(3^2 + 2^2 + (-5)^2)`

`|bar v| = sqrt 38`

You may evaluate the cosine of th angle between the vectors `bar u ` and `bar v` , such that:

`cos(hat(bar u, bar v)) = (bar u*bar v)/(|bar u|*|bar v|)`

`cos(hat(bar u, bar v)) = 24/(sqrt(21*38)) => cos(hat(bar u, bar v)) = 24/(sqrt798) => cos(hat(bar u, bar v)) = 24/28.24 = 0.849 => (hat(bar u, bar v)) ~~ 32^o` .

**Hence, evaluating the angle between the given vectors yields that `(hat(bar u, bar v)) ~~ 32^o` , thus `(hat(bar u, bar v)) ~~ 32^o` is an acute angle.**