Scalar product of vectors (hereinafter referred to as SP). Dear friends! The mathematics exam includes a group of problems on solving vectors. We have already considered some problems. You can see them in the “Vectors” category. In general, the theory of vectors is not complicated, the main thing is to study it consistently. Calculations and operations with vectors in school course The math is simple, the formulas are not complicated. Take a look at. In this article we will analyze problems on SP of vectors (included in the Unified State Examination). Now “immersion” in the theory:
H To find the coordinates of a vector, you need to subtract from the coordinates of its endthe corresponding coordinates of its origin
And further:
*Vector length (modulus) is determined as follows:
These formulas must be remembered!!!
Let's show the angle between the vectors:
It is clear that it can vary from 0 to 180 0(or in radians from 0 to Pi).
We can draw some conclusions about the sign dot product. The lengths of vectors have a positive value, this is obvious. This means the sign of the scalar product depends on the value of the cosine of the angle between the vectors.
Possible cases:
1. If the angle between the vectors is acute (from 0 0 to 90 0), then the cosine of the angle will have a positive value.
2. If the angle between the vectors is obtuse (from 90 0 to 180 0), then the cosine of the angle will have a negative value.
*At zero degrees, that is, when the vectors have the same direction, the cosine is equal to one and, accordingly, the result will be positive.
At 180 o, that is, when the vectors have opposite directions, the cosine is equal to minus one,and accordingly the result will be negative.
Now the IMPORTANT POINT!
At 90 o, that is, when the vectors are perpendicular to each other, the cosine is equal to zero, and therefore the SP is equal to zero. This fact (consequence, conclusion) is used in solving many problems where we are talking about relative position vectors, including in problems included in open bank math assignments.
Let us formulate the statement: the scalar product is equal to zero if and only if these vectors lie on perpendicular lines.
So, the formulas for SP vectors:
If the coordinates of the vectors or the coordinates of the points of their beginnings and ends are known, then we can always find the angle between the vectors:
Let's consider the tasks:
27724 Find the scalar product of the vectors a and b.
We can find the scalar product of vectors using one of two formulas:
The angle between the vectors is unknown, but we can easily find the coordinates of the vectors and then use the first formula. Since the origins of both vectors coincide with the origin of coordinates, the coordinates of these vectors are equal to the coordinates of their ends, that is
How to find the coordinates of a vector is described in.
We calculate:
Answer: 40
Let's find the coordinates of the vectors and use the formula:
To find the coordinates of a vector, it is necessary to subtract the corresponding coordinates of its beginning from the coordinates of the end of the vector, which means
We calculate the scalar product:
Answer: 40
Find the angle between vectors a and b. Give your answer in degrees.
Let the coordinates of the vectors have the form:
To find the angle between vectors, we use the formula for the scalar product of vectors:
Cosine of the angle between vectors:
Hence:
The coordinates of these vectors are equal:
Let's substitute them into the formula:
The angle between the vectors is 45 degrees.
Answer: 45
Example 1.
Find the scalar product of the vectors a = (1; 2) and b = (4; 8).
Solution: a · b = 1 · 4 + 2 · 8 = 4 + 16 = 20.
Example 2.
Find the scalar product of vectors a and b if their lengths |a| = 3, |b| = 6, and the angle between the vectors is 60˚.
Solution: a · b = |a| · |b| cos α = 3 · 6 · cos 60˚ = 9.
Example 3.
Find the scalar product of the vectors p = a + 3b and q = 5a - 3 b if their lengths |a| = 3, |b| = 2, and the angle between vectors a and b is 60˚.
Solution:
p · q = (a + 3b) · (5a - 3b) = 5 a · a - 3 a · b + 15 b · a - 9 b · b = = 5 |a| 2 + 12 a · b - 9 |b| 2 = 5 3 2 + 12 3 2 cos 60˚ - 9 2 2 = 45 +36 -36 = 45.
An example of calculating the scalar product of vectors for spatial problems
Example 4.
Find the scalar product of the vectors a = (1; 2; -5) and b = (4; 8; 1).
Solution: a · b = 1 · 4 + 2 · 8 + (-5) · 1 = 4 + 16 - 5 = 15.
An example of calculating the dot product for n-dimensional vectors
Example 5.
Find the scalar product of the vectors a = (1; 2; -5; 2) and b = (4; 8; 1; -2).
Solution: a · b = 1 · 4 + 2 · 8 + (-5) · 1 + 2 · (-2) = 4 + 16 - 5 -4 = 11.
Vector addition of vectors, power. Geometrical and physical displacement. Calculation of the vector addition based on the known coordinates of multiplying vectors.
Cross product of vectors and its properties
The vector is called vector product non-collinear vectors and if:
1) its length is equal to the product of the lengths of the vectors and the sine of the angle between them: (Fig. 1.42);
2) the vector is orthogonal to the vectors and ;
3) vectors , , (in the indicated order) form a right triple.
The vector product of collinear vectors (in particular, if at least one of the factors is a zero vector) is considered equal to the zero vector.
The cross product is denoted by (or ).
Algebraic properties of a vector product
For any vectors , , and any real number:
1. 
;
3. 
.
The first property determines the antisymmetry of the vector product, the second and third - additivity and homogeneity with respect to the first factor. These properties are similar to the properties of the product of numbers: the first property is “opposite” to the law of commutativity of multiplication of numbers (the law of anticommutativity), the second property corresponds to the law of distributivity of multiplication of numbers in relation to addition, the third - the law of associative multiplication. Therefore, the operation under consideration is called the product of vectors. Since its result is a vector, such a product of vectors is called a vector product.
Let us prove the first property, assuming that the vectors and are not collinear (otherwise both sides of the equality being proved are equal to the zero vector). By definition, the vectors and have equal lengths and are collinear (since both vectors are perpendicular to the same plane). By definition, triples of vectors and are right-handed, i.e. the vector is directed so that the shortest turn from k occurs in the positive direction (counterclockwise), when viewed from the end of the vector, and the vector is directed so that the shortest turn from k occurs in the positive direction, when viewed from the end of the vector (Fig. 1.43) . This means that the vectors and are in opposite directions. Therefore, that is what needed to be proven. The proof of the remaining properties is given below (see paragraph 1 of remarks 1.13).
Dot product of vectors.Dot product of vectors. Online calculators for scalar product and angle between vectors by coordinates.
The dot product of vectors is an operation on two vectors that results in a number (not a vector).
Determined The dot product is typically as follows:
In other words, the scalar product of vectors is equal to the product of the lengths of these vectors and the cosine of the angle between them. It should be noted that the angle between two vectors is the angle they form if they are set aside from one point, that is, the origins of the vectors must coincide.
The following simple properties follow directly from the definition:
1. The scalar product of an arbitrary vector a and itself (scalar square of vector a) is always non-negative, and equal to the square of the length of this vector. Moreover, the scalar square of a vector is equal to zero if and only if given vector- zero.
2. The scalar product of any perpendicular vectors a and b is equal to zero.
3. The scalar product of two vectors is zero if and only if they are perpendicular or at least one of them is zero.
4 . The scalar product of two vectors a and b is positive if and only if there is an acute angle between them.
5. The scalar product of two vectors a and b is negative if and only if there is an obtuse angle between them.
An alternative definition of the dot product, or calculating the dot product of two vectors given their coordinates.
(Calculating the coordinates of a vector if the coordinates of its beginning and end are given is very simple -
Let there be a vector AB, A - the beginning of the vector, B - the end, and the coordinates of these points
A=(a 1,a 2,a 3), B=(b 1,b 2,b 3)
Then the coordinates of the vector AB are:
AB=(b 1 -a 1, b 2 -a 2, b 3 -a 3).
Similarly in two-dimensional space - there are simply no third coordinates)
So, let two vectors be given, defined by a set of their coordinates:
a) In two-dimensional space (on a plane).
Then their scalar product can be calculated using the formula:
b) In three-dimensional space
Similar to the two-dimensional case, their scalar product is calculated using the formula:
Calculate the angle between vectors using the dot product.
The most common mathematical application of the dot product of two vectors is to calculate the angle between vectors given their coordinates. Let's take the three-dimensional case as an example. (If the vectors are specified on the plane, that is, by two coordinates, all formulas simply lack third coordinates.)
So let's say we have two vectors:
And we need to find the angle between them. Using their coordinates, we find their lengths, and then simply equate the two formulas for the scalar product. This way we get the cosine of the desired angle.
The length of the vector a is calculated as the root of scalar square vector a, which we calculate using the formula for the scalar product of vectors specified by coordinates:
The length of the vector b is calculated similarly.
The required angle has been found.
Online calculator for the scalar product of two vectors.
To find the scalar product of two vectors using this calculator, you need to enter in the first line in order the coordinates of the first vector, in second - second. The coordinates of vectors can be calculated from the coordinates of their start and end (see above for an alternative definition of the dot product, or calculating the dot product of two vectors given their coordinates.)
If vectors are specified by two coordinates, then the third coordinate of each vector must be filled with a zero.