Instruction

1

The angle between two nonzero vectors is determined by calculating the scalar product. By definition, the scalar product equals the product of the lengths of vectors on a cosine of the angle between them. On the other hand, the scalar product for two vectors a with coordinates (x1; y1) and b with coordinates (x2; y2) is calculated according to the formula: ab = x1x2 + y1y2. Of these two methods of finding the scalar product it is easy to find the angle between the vectors.

2

Find the length or the modules of the vectors. For our vectors a and b: |a| = (x12 + y12)^1/2, |b| = (x22 + y22)^1/2.

3

Find the scalar product of vectors, multiplying their coordinates by pairs: ab = x1x2 + y1y2. From the definition of the scalar product ab = |a|*|b|*cos α, where α is the angle between the vectors. Then we get that x1x2 + y1y2 = |a|*|b|*cos α. Then cos α = (x1x2 + y1y2)/(|a|*|b|) = (x1x2 + y1y2)/((x12 + y12)(x22 + y22))^1/2.

4

Find the angle α using the tables Bradis.

5

In the case of three-dimensional space adds a third coordinate. For vectors a (x1; y1; z1) and b (x2; y2; z2), the formula for the cosine of the angle is shown in Fig.

Note

The scalar product is a scalar characteristic of the lengths of vectors and angle between them.

# Advice 2 : How to find the angle between two vectors

**The angle**between the two

**vectors**, coming from the same point, this is the shortest angle to rotate one of the vectors around their start to the second vector. Determine the degree measure of this angle is possible, if you know the coordinates of the vectors.

Instruction

1

Let on the plane contains two nonzero vectors, delayed from one point: the vector A with coordinates (x1, y1) and vector B with coordinates (x2, y2).

**The angle**between them is designated as θ. To find the degree measure of the angle θ it is necessary to use the definition of the scalar product.2

Scalar product of two nonzero vectors is a number equal to the product of the lengths of these vectors on a cosine of the angle between them, i.e. (A,B)=|A|*|B|*cos(θ). Now we need to Express from this record is the cosine of the angle: cos(θ)=(A,B)/(|A|*|B|).

3

The scalar product can also be found by the formula (A,B)=x1*x2+y1*y2, i.e. the scalar product of two nonzero vectors is equal to the sum of the products of the corresponding coordinates of those vectors. If the scalar product of nonzero vectors is zero, the vectors are perpendicular (the angle between them is 90 degrees) and further computation can produce. If the scalar product of two vectors is positive, then the angle between these

**vectors**is acute, and if negative, the angle is obtuse.4

Now calculate the lengths of vectors A and B by the formulas: |A|=√(x12+y12), |B|=√(x22+y22). The length of the vector is calculated as the square root of the sum of the squares of its coordinates.

5

The values of the scalar product and lengths of vectors substitute in obtained in step 2 the formula for finding the cosine of the angle, that is cos(θ)=(x1*x2+y1*y2)/(√(x12+y12)+√(x22+y22)). Now that we know the cosine ratio to find the degree measure of the angle between the

**vectors**need to use the table Bradis or take from this expression the arc cosine is: θ=arccos(cos(θ)).6

If the vectors A and B given in three-dimensional space have coordinates (x1, y1, z1) and (x2, y2, z2), respectively, when the cosine angle is another coordinate. In this case, the cosine of the angle is: cos(θ)=(x1*x2+y1*y2+z1*z2)/(√(x12+y12+z12)+√(x22+y22+z22)).

Useful advice

If the two vectors are not deferred from one point, to find the angle between a parallel transfer need to combine the start of these vectors.

The angle between the two vectors cannot be greater than 180 degrees.

The angle between the two vectors cannot be greater than 180 degrees.