Determine [3a-4b].[2a+b], given a and b are unit vectors and the angle between them is 30 degree.  

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to determine [3a-4b].[2a+b], given that a and b are unit vectors and the angle between them is 30 degrees.

a dot b = |a||b| cos 30 = sqrt 3/2

[3a-4b].[2a+b]

=> 3a dot 2a + 3a dot b - 4b dot 2a - 4b dot b

=> 6*a dot a + 3a dot b - 8* a dot b - 4* b dot b

=> 6 - 5(a dot b) - 4

=> 2 - 5*sqrt 3/2

=> (4 - 5*sqrt 3)/2

The required result is (4 - 5*sqrt 3)/2

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll determine the dot product of a and b:

a*b = |a|*|b|*cos 30 = 1*1*sqrt3/2

We'll use FOIL method to remove the brackets:

(3a-4b)(2a+b) = 3a^2 + 3ab - 8ab - 4b^2

The angle made by the vector a with itself is of 0 degrees.

a^2 = a*a = |a|*|a|*cos 0 = 1*1*1 = 1

b^2 = b*b = 1

(3a-4b)(2a+b) = 6 + 3*sqrt3/2 - 8sqrt3/2 - 4

We'll combine like terms:

(3a-4b)(2a+b) = 2 - 5sqrt3/2

The requested value of the product is (3a-4b)(2a+b) = 2 - 5sqrt3/2.

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