The definition and basic properties of vector inner product, 1. Given vector a = (1, radical 3), vector b = (- radical 3, - 1), find 2. Given points a (x, - 1), B (- 2, - 6), C (1, - 2) and | vector ab | = | vector AC |, find the value of X 3. Given points a (x, 4), B (2, y + 3), and vector AB = (3, 6), find the value of X, y | Math enthusiast | Mathematical art

The definition and basic properties of vector inner product, 1. Given vector a = (1, radical 3), vector b = (- radical 3, - 1), find 2. Given points a (x, - 1), B (- 2, - 6), C (1, - 2) and | vector ab | = | vector AC |, find the value of X 3. Given points a (x, 4), B (2, y + 3), and vector AB = (3, 6), find the value of X, y

1.a = (1,√3),b = (-√3,-1)
cos = a•b /∣a∣∣b∣
= [1 * (-√3) + √3 * (-1)] / √[1² + (√3)²] * √[(-√3)² + (-1)²]
= -2√3 / 4
= -√3 / 2
So = 180 ° - 30 ° = 150 °
2.AB = (-2,-6) - (x,-1)
= (-2-x,-6+1)
= (-2-x,-5)
AC = (1,-2) - (x,-1)
= (1-x,-2+1)
= (1-x,-1)
And ∣ ab ∣ = ∣ AC ∣
√[(-2-x)² + (-5)²] = √[(1-x)² + (-1)²]
Both sides square at the same time, X & # 178; + 4x + 4 + 25 = x & # 178; - 2x + 1 + 1
The solution is x = - 9 / 2
3.AB = (2,y+3) - (x,4)
= (2-x,y+3-4)
= (2-x,y-1)
And ab = (3,6)
SO 2 - x = 3, Y - 1 = 6
The solution is x = - 1, y = 7

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