Vectors. Triangle Angle between vector

In summary, we are given three points on the Cartesian plane (A(1,2), B(-2,-1), and C(3,-2)). We are asked to find the length of segments AB and BC, as well as the angle of the triangle at vertex B. To find the length of these segments, we use the formula |AB| = √[(x2-x1)^2 + (y2-y1)^2]. Plugging in the values, we get |AB| = √(18) and |BC| = √(26). To find the angle, we can use the formula cosθ = (AB*BC)/(|AB|*|BC|). However, the dot product should
  • #1
SpringWater
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Homework Statement


Consider the points A(1,2) B(-2,-1) and C(3,-2) on the Cartesian plane.

1. Find the length of the segments AB and BC

2. Find the angle of the triangle at the vertex B


Homework Equations



Cosθ=(AB*BC)/|AB| |BC| ?

The Attempt at a Solution



The length of the segments; AB= <-3,-3> |AB|=√(18)
BC =<5,-1> |BC|=√(26)

So therefore the angle at point B is θ=arccos(((-3)(5)) (dot) ((-1)(-3))) / (√(18*26))

θ=arccos(-18 / √(468))

is this correct?

I am assuming that the dot product would be the (x1)(x2) + (y1)(y2)
 
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  • #2
The idea looks right, but (-3)(5)+(-1)(-3) isn't -18.
 

Related to Vectors. Triangle Angle between vector

What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is often represented graphically as an arrow, with the length of the arrow representing the magnitude and the direction of the arrow indicating the direction of the vector.

How is the angle between two vectors measured?

The angle between two vectors is measured using the dot product formula: θ = cos⁻¹((u · v) / (|u| * |v|)), where u and v are the two vectors and |u| and |v| are the magnitudes of the vectors. This formula gives the angle in radians, so if you want the angle in degrees, you can multiply the result by 180/π.

What does the angle between two vectors represent?

The angle between two vectors represents the amount of rotation needed to align one vector with the other. If the angle is 0 degrees, the vectors are pointing in the same direction. If the angle is 180 degrees, the vectors are pointing in opposite directions. Any angle in between represents a degree of rotation.

What is the relationship between the angle between two vectors and their dot product?

The dot product of two vectors is equal to the product of their magnitudes multiplied by the cosine of the angle between them. This means that the dot product can be used to calculate the angle between two vectors, and vice versa.

How do you find the angle between two vectors in three-dimensional space?

In three-dimensional space, the angle between two vectors can be found using the same dot product formula as in two-dimensional space, but with an additional term for the third component of each vector. The formula is: θ = cos⁻¹((u · v) / (|u| * |v| * cos(ϕ)), where ϕ is the angle between the projections of the two vectors on the xy-plane.

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