Two vectors and the angle between them

Mathematicians and physicists often need to find the angle between two given vectors. While it is easy to find the angle between two vectors in the same plane by drawing a sketch, finding the angle between 3D vectors can be slightly trickier. This article details the method to find out the angle between two vectors, whether in two dimensions or three.

Show Ads

[edit] Steps

Use OM and OQ

Determine the vectors you must use to find the angle. Say two vectors OM and OQ intersect at point O, and you need to calculate the angle MOQ. You must use vectors OM and OQ, not MO or QO. If you know MO, multiply it by -1 (negative one) to give OM and use that.
Find the scalar product (or dot product) of the two vectors. If you do not know how to calculate the scalar product between two vectors, here's how:
1. Identify the components of the vector in each direction. If the vector is given as a column vector, the first row usually represents the x-axis, the second row the y-axis, and the third row the z-axis. If the vector is given in the form xi + yj + zk, the coefficients of i,j, and k represent the magnitudes of the components along the x-, y-, and z-axes respectively (i,j, and k are unit vectors along the x-, y-, and z-axes respectively).
2. Determination of scalar products. The same two vectors represented in i,j,k and column form
  
  Multiply the components of both vectors along the x-axis with each other. Then multiply the components of both vectors along the y-axis with each other, and do the same for the components along the z-axis.
3. Add the three multiplication products together. This is the scalar product of the two vectors. It does not "represent" anything, but its sole purpose is to aid the calculation of the angle between two vectors. In a two dimensional vector, the component along the z-axis is zero, so the scalar product is found by considering the components along the x- and y-axes only.
Calculate the magnitude of the two vectors using the formula a²=b²+c²+d², where a is the magnitude of the vector, and b,c, and d are the magnitudes of the components in the three directions. In a two dimensional vector, d will equal zero.
The scalar product (often represented by a.b) equals |a||b|cosθ, where |a| is the magnitude of one vector, |b| the magnitude of the other, and θ the angle between the two. If the scalar product is negative, use the negative value.
- If you need the acute angle between the two two vectors and the above method yields an obtuse angle, subtract if from 180^o