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Decomposition of vectors.Theorem 1. An arbitrary vector
Theorem 2. An arbitrary vector
Let The Cartesian system of coordinates. Consider the following coordinate system: take mutually perpendicular unit vectors
Definition. The triple of vectors From the triangle ÎÌÌ1, we obtain
Since the vector
From the triangle ÎÀÌ1, we obtain
because, by analogy, the vectors
Thus, the radius vector Consider the vectors
and their sum
Under addition the respective coordinates are added Let us multiply the vector
When a vector is multiplied by a number l, each coordinate of this vector is multiplied by this number. Example. Find the vector
Let us find the required vector in vector notation:
To find the same vector in vector notation, we multiply the first vector by 4 and the second by –3 and sum their coordinates:
Given two points Thus, we have found the required vector in the coordinate notation:
To find the coordinates of a vector, we must subtract the coordinates of its tail from the coordinates of the head. For example, let us find vectors with given coordinates of heads and tails: Ì1(7;4;–3); Ì2(1;–2;–2);
Find the length of a vector
From the right triangle ÎÌ1Ì2 , we find the hypotenuse
where M1 From the other right triangle ÎÀÌ2 , 0 z1 y we find the hypotenuse Substituting it into À x1 the first hypotenuse, we obtain x y1 M2
Thus, the length of a vector is defined by the formula
Date: 2015-01-02; view: 1692
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