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How to calculate the angle between two 3D vectors

Calculate the angle between two intersecting unit vectors

If we wish to find the angle created by 3 points P, Q and S we must derived the unit vectors PQ and RS

		Q_x - P_x
PQ =		Q_y - P_y
		Q_z - P_z

		S_x - P_x
PS =		S_y - P_y
		S_z - P_z

Convert to unit vectors:

PQ length, PQ_l = √( PQ_x² + PQ_y² + PQ_z² )

		PQ_x / PQ_l
Convert PQ to a unit vector =		PQ_y / PQ_l
		PQ_z / PQ_l

PS length, PS_l = √( PS_x² + PS_y² + PS_z² )

		PS_x / PS_l
Convert PS to a unit vector =		PS_y / PS_l
		PS_z / PS_l

Now we have the 2 unit vectors PQ and PS.

The angle α between PQ and PS:

α = cos^-1(PQ.PS)

PQ.PS = ( PQ_x × PS_x ) + ( PQ_y × PS_y ) + ( PQ_z × PS_z )

∴ α = cos^-1( ( PQ_x × PS_x ) + ( PQ_y × PS_y ) + ( PQ_z × PS_z ) )

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