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.

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

A vector is a segment with the given direction. The angle between the vectors has a physical meaning, for example when finding the length of the projection of the vector on the axis.

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 3: How to find the cosine of the angle between the vectors

Vector in geometry is called a directed line segment or an ordered pair of points in Euclidean space. The length of the vector is a scalar equal to the arithmetic square root of the sum of the squares of the coordinate (component) of the vector.

You will need

- A basic knowledge of geometry and algebra.

Instruction

1

The cosine of the angle between the vectors, find their dot product. The sum of the products of the respective coordinates of the vector equal to the product of their lengths into the cosine of the angle between them. Let the given two vectors: a(x1, y1) and b(x2, y2). Then the scalar product can be written as the equation: x1*x2 + y1*y2 = |a|*|b|*cos(U), where U is the angle between the vectors.

For example, the coordinates of the vector a(0, 3) and vector b(3, 4).

For example, the coordinates of the vector a(0, 3) and vector b(3, 4).

2

Expressing the equality cos(U) it turns out that cos(U) = (x1*x2 + y1*y2)/(|a|*|b|). In the example, the formula after substitution of known coordinates takes the form: cos(U) = (0*3 + 3*4)/(|a|*|b|) or cos(U) = 12/(|a|*|b|).

3

The length of the vectors is according to the formula: |a| = (x1^2 + y1^2)^1/2, |b| = (x2^2 + y2^2)^1/2. Substituting the coordinates of the vectors a(0, 3), b(3, 4) is obtained, respectively, |a|=3, |b|=5.

4

Substituting these values into the formula cos(U) = (x1*x2 + y1*y2)/(|a|*|b|), find the answer. Found using the lengths of vectors, we find that the cosine of the angle between the vectors a(0, 3), b(3, 4) is equal to: cos(U) = 12/15.

Note

If everything is calculated correctly, the cosine of the angle must be less than unity. Also the lengths of the vectors cannot be a negative number.

Useful advice

If the length of one of the vectors is zero, then it is zero vector, then the angle between it and another vector equal to 90 degrees.

# Advice 4: How to find the degree measure of angle

Measure flat of angles in degrees was invented in ancient Babylon long before our era. The people of the state preferred a sexagesimal system of calculation, so division corners on 180 or 360 units today looks a little weird. However, proposed in the modern system of SI units, multiples of the number PI, not less weird. These two options are not limited to the currently used designations of angles, so the task of translating their values in degree measure arises quite often.

Instruction

1

If the degree measure of the need to translate the measure of the angle in radians, assume that one degree corresponds to the number of radians equal to 1/180 fraction of PI. This mathematical constant has an infinite number of decimal places, so the conversion factor from radians to degrees is also an infinite decimal fraction. This means that an absolutely exact value in decimal fractions to not work, so the conversion factor should be rounded to. For example, with a precision of a billionth fraction of a unit calculated ratio is equal to 0,017453293. After rounding to the desired number of characters, divide by this factor the initial number of radians and you will get a degree angle.

2

In solving mathematical tasks from the topics that are related to geometry, often there are formulas in which values of angles are radians and fractions of PI. If you get a solution that contains this constant, to convert it to degrees, replace π by the number 180. For example, if the Central angle is defined by the expression π/4, this means that its degree measure is equal to 180°/4=45°.

3

Angles can be expressed in units, which are called "revolution." This unit corresponds to 360°, so the problems with conversion should not occur. For example, if the task says about the angle in a turn and a half, this corresponds to 360 x 1.5=540° in degree measure.

4

Sometimes in geometry problems is referred to an obtuse angle. It is formed by two rays, called opposite direction, that is, collinear. Use the number 180 for the expression values of the expanded angle in degrees.

5

In geodesy, cartography, astronomy degrees are divided into still smaller units, which have their own names, minutes and seconds. This division has roots in the same place and degrees, so each degree comprises 60 minutes, or 3600 seconds. Use these numbers if seconds and minutes need to be replaced in tenths of a degree. For example, angle 11°14'22" corresponds to a decimal fraction, approximately equal to 11 + 14/60 + 22/3600 if 11,2394°.