# The vectors a and b have lengths 2 and 1, respectively. The vectors a+5b and 2a-3b are perpendicular. Determine the angle between a and b. You need to write the vectors `bar a` and `bar b` , such that:

`bar a = a_1*bar i + a_2*bar j`

`bar b = b_1*bar i + b_2*bar j`

The problem provides the information that the vectors `bar a + 5bar b` and `2bar a - 3bar b` are...

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You need to write the vectors `bar a` and `bar b` , such that:

`bar a = a_1*bar i + a_2*bar j`

`bar b = b_1*bar i + b_2*bar j`

The problem provides the information that the vectors `bar a + 5bar b` and `2bar a - 3bar b` are perpendicular, hence, its dot product yields 0, such that:

`(bar a + 5bar b)(2bar a - 3bar b) = 0`

`(a_1*bar i + a_2*bar j + 5b_1*bar i + 5b_2*bar j)(2a_1*bar i + 2a_2*bar j - 3b_1*bar i - 3b_2*bar j) = 0`

`((a_1+5b_1)*bar i + (a_2+5b_2)*bar j)((2a_1 - 3b_1)*bar i + (2a_2 - 3b_2)*bar j) = 0`

`(a_1+5b_1)(2a_1 - 3b_1) +(a_2+5b_2)(2a_2 - 3b_2) = 0`

`2a_1^2 + 7a_1*b_1 - 15b_1^2 + 2a_2^2 + 7a_2*b_2 - 15b_2^2 = 0`

`7(a_1*b_1 + a_2*b_2) = 15(b_1^2 + b_2^2) - 2(a_1^2 + a_2^2)`

`(a_1*b_1 + a_2*b_2) = (15(b_1^2 + b_2^2) - 2(a_1^2 + a_2^2))/7`

You need to determine the angle between the vectors `bar a` and `bar b` , hence, you need to evaluate its dot product, such that:

`bar a*bar b = |bar a|*|bar b|*cos(hat(bar a,bar b))`

Since the problem provides the absolute values of `bar a` and `bar b` yields:

`bar a*bar b =2*1*cos(hat(bar a,bar b)) =>cos(hat(bar a,bar b))= (bar a*bar b)/2`

Evaluating the product `bar a*bar b` yields:

`bar a*bar b = a_1*b_1 + a_2*b_2`

Replacing `(15(b_1^2 + b_2^2) - 2(a_1^2 + a_2^2))/7` for `a_1*b_1 + a_2*b_2` yields:

`cos(hat(bar a,bar b)) = (15(b_1^2 + b_2^2) - 2(a_1^2 + a_2^2))/14`

`hat(bar a,bar b) = cos^(-1)((15(b_1^2 + b_2^2) - 2(a_1^2 + a_2^2))/14)`

Hence, evaluating the angle between the vectros `bar a` and `bar b` , yields `hat(bar a,bar b) = cos^(-1)((15(b_1^2 + b_2^2) - 2(a_1^2 + a_2^2))/14).`

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