Aijk 4 42 and ˆ 42 4 − are two vectors the angle between them will betheanglebetweenthemwillbe

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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

1. 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))]

1. 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

1. 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))]

1. 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.