The length of the directed segment determines the numerical value of the vector and is called vector length or vector module. To indicate the magnitude of the vector, two vertical lines are used on the left and right |AB|.

Vector modulus (vector length)|a| in rectangular Cartesian coordinates is equal to the square root of the sum of the squares of its coordinates.

So, in the case of a plane problem, the vector modulus can be found using the following formula
|(a)| = sqrt(x_1^2+y_1^2).

Example
Find the length of the vector (a) = (2;4).
Solution:|(a)| = sqrt(2^2+4^2)=sqrt(4+16)=sqrt(20)=2sqrt(5).

So, in the case of a spatial problem, the modulus of the vector (a) = (x_1;y_1;z_1) can be found using the following formula |(a)| = sqrt(x_1^2+y_1^2+z_1^2).

Examplecalculating vector modulus (vector length)
Find the length of the vector (a) = (2; 4; 4).
Solution:|(a)| = sqrt(2^2+4^2+4^2)=sqrt(4+16+16)=sqrt(36)=6.

Orthogonal vectors. Orthonormal basis.

Definition. Two vectors are called orthogonal if the angle between them is equal to a right angle, i.e. .

Designation: – vectors and orthogonal.

Definition. A triple of vectors is called orthogonal if these vectors are pairwise orthogonal to each other, i.e. , .

Definition. A triple of vectors is called orthonormal if it is orthogonal and the lengths of all vectors are equal to one: .

Comment. From the definition it follows that an orthogonal and, therefore, orthonormal triple of vectors is non-coplanar.

Definition. An ordered non-coplanar triple of vectors plotted from one point is called right (right-oriented) if, when observing from the end of the third vector to the plane in which the first two vectors and lie, the shortest rotation of the first vector to the second occurs counterclockwise. Otherwise, the triple of vectors is called left (left-oriented).

Here, in Fig. 6, the right three of vectors is shown. The following Fig. 7 shows the left triple of vectors:

Definition. A basis of a vector space is said to be orthonormal if a triple of vectors is orthonormal.

Designation. In what follows we will use a right orthonormal basis, see the following figure:

vector module- vector magnitude - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information Technology in general Synonyms value of a vector EN absolute value of a vector ...

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Vector module can be found if we know it projections onto coordinate axes.

given on the plane vector A(Fig. 15).

Let us drop perpendiculars from the beginning and end of the vector onto the coordinate axes to find its projections. In accordance with the Pythagorean theorem

. From here

.

You need to know this formula BY HEART.

Remember!

To find vector module it is necessary to extract the square root of the sum of the squares of its projections.

You already know that the projection of a vector onto an axis can be found if you subtract the coordinate of its beginning point from the coordinate of the end point of the vector. Then for our vector, if it is given on the plane, and x = x k − x n,
and y = y k − y n. Hence, vector module can be found using the formula

.

It is not difficult to imagine what the formula will look like if vector given in space.

Also pay attention to this. After all vector module is the length of the segment enclosed between two points: the beginning point of the vector and the end point. And this is nothing more than the distance between these two points. Therefore, to find the distance between any two points, you need to calculate vector module connecting these points.