In this paragraph, you'll find the formulas for the angle between two vectors - and only the formulas. If you'd like to understand how we derive them, go directly into the next paragraph, How to find the angle between two vectors
Angle between two 2D vectors
- Vectors represented by coordinates (standard ordered set notation, component form):
vectors a = [xa, ya] , b = [xb, yb]
angle = arccos[(xa * xb + ya * yb) / (√(xa2 + ya2) * √(xb2 + yb2))]
- Vectors between a starting and terminal point:
For vector a: A = [x1, y1] , B = [x2, y2],
so vector a = [x2 - x1, y2 - y1]
For vector b: C = [x3, y3] , D = [x4, y4],
so vector b = [x4 - x3, y4 - y3]
Then insert the derived vector coordinates into the angle between two vectors formula for coordinate from point 1:
angle = arccos[((x2 - x1) * (x4 - x3) + (y2 - y1) * (y4 - y3)) / (√((x2 - x1)2 + (y2 - y1)2) * √((x4 - x3)2 + (y4 - y3)2))]
Angle between two 3D vectors
- Vectors represented by coordinates:
a = [xa, ya, za] , b = [xb, yb, zb]
angle = arccos[(xa * xb + ya * yb + za * zb) / (√(xa2 + ya2 + za2) * √(xb2 + yb2 + zb2))]
- Vectors between a starting and terminal point:
For vector a: A = [x1, y1, z1], B = [x2, y2, z2],
so a = [x2 - x1, y2 - y1, z2 - z1]
For vector b: C = [x3, y3, z3], D = [x4, y4, z4]
so b = [x4 - x3, y4 - y3, z4 - z3]
Find the final formula analogically to the 2D version:
angle = arccos{[(x2 - x1) * (x4 - x3) + (y2 - y1) * (y4 - y3) + (z2 - z1) * (z4 - z3)] / [√((x2 - x1)2 + (y2 - y1)2+ (z2 - z1)2) * √((x4 - x3)2 + (y4 - y3)2 + (z4 - z3)2)]}
Also, it is possible to have one angle defined by coordinates, and the other defined by a starting and terminal point, but we won't let that obscure this section even further. All that matters is that our angle between two vectors calculator has all possible combinations available to you.
Intersection of two lines
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[1] 2021/05/06 20:04 40 years old level / A teacher / A researcher / Very /
Purpose of useTo calculate distance measurements in architectural drawings. Comment/RequestI find that using this calculator site works better than the others I have tried for finding the equations and intersections of lines.I wish that it would graph these solutions though.
[2] 2021/05/03 01:52 40 years old level / An engineer / Useful /
Purpose of usequicker than calculating optimizations mentally[3] 2021/02/04 23:41 Under 20 years old / Elementary school/ Junior high-school student / A little /
Purpose of usemath test[4] 2020/02/25 23:24 20 years old level / An office worker / A public employee / Very /
Purpose of useFigured out my employer's new payment calculation isn't actually shortchanging me.[5] 2014/10/31 19:28 20 years old level / High-school/ University/ Grad student / Very /
Purpose of useto help to solve a solution that i cant find where i am going wrong[6] 2014/05/06 22:29 Under 20 years old / Elementary school/ Junior high-school student / Not at All /
Purpose of usehw[7] 2013/09/30 21:13 60 years old level or over / Others / Very /
Purpose of usePuzzle solutionComment/Requestvery good, thank you[8] 2013/04/03 18:07 Under 20 years old / Elementary school/ Junior high-school student / A little /
Purpose of useto check a program i am writing[9] 2013/03/25 22:00 Under 20 years old / High-school/ University/ Grad student / Very /
Purpose of useto determine an unclear point of intersection.[10] 2009/11/24 19:59 More than 60 / A researcher / Very /
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