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.      The vectors a and b have lengths 2 and 1,...

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.

 

 

 

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.

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should come up with the notations for the vectors `bar a+5 bar b` and `2bar a - 3bar b` such that:

`bar u = bar a + 5 bar b = (x_a*bar i + y_a*bar j) + 5(x_b*bar i + y_b bar j)`

`bar v = 2bar a - 3bar b = 2(x_a*bar i + y_a*bar j) - 3(x_b*bar i + y_b bar j)`

Since the problme provides the information that the vectors `bar u` and `bar v` are perpendicular to each other, hence the dot product is zero, such that:

`bar u*bar v = 0`

`((x_a*bar i + y_a*bar j) + 5(x_b*bar i + y_b bar j))(2(x_a*bar i + y_a*bar j) - 3(x_b*bar i + y_b bar j)) = 0`
 => `{((x_a*bar i + y_a*bar j) + 5(x_b*bar i + y_b bar j) = 0),(2(x_a*bar i + y_a*bar j) - 3(x_b*bar i + y_b bar j) = 0):}` `=> {((x_a*bar i + y_a*bar j) = -5(x_b*bar i + y_b bar j) ),(-10x_b bar i - 10y_b bar j - 3x_b bar i - 3y_b bar j = 0):}` `=> bar b = -13x_b bar i - 13y_b bar j`

`bar a = 15x_a bar i + 15y_a bar j`

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

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

Since the problem provides the magnitudes of the vectors ` bar a `  and `bar b` , yields:

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

You need to perform the multiplication of vectors `bar a` and `bar b` , such that:

`bar a*bar b = (15x_a bar i + 15y_a bar j)(-13x_b bar i - 13y_b bar j) = -195x_a*x_b - 195y_a*y_b`

`-195x_a*x_b - 195y_a*y_b = 2*cos hat((bar a,bar b))`

`cos hat((bar a,bar b)) = (-195(x_a*x_b + y_a*y_b))/2`

You need to remember that the values of cosine are located in interval `[-1,1]` such that:

`-1 <= (-195(x_a*x_b + y_a*y_b))/2 =< 1`

`2/195 >= x_a*x_b + y_a*y_b >= -2/195`

Hence, evaluating the angle between the vectors ` bar a` and `bar b` , yields `cos hat((bar a,bar b)) = (-195(x_a*x_b + y_a*y_b))/2` , if `2/195 >= x_a*x_b + y_a*y_b >= -2/195.`

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