The position vector refers to the coordinateoriginA directed line segment starting from the position of the moving particle and ending at the position of the moving particle.displacementAlthough both and vectors are vectors, they are two different concepts.The position vector istime, a directed line segment starting from the coordinate origin and ending at the position of the moving particle;The displacement is a directed line segment leading from the starting position of the particle to the ending position of the particle in a period of time interval.
The position vector refers to a directed line segment starting from the coordinate origin and ending at the position of the moving particle at a certain time.
Affine geometrydefinition:A is an affine space, if any point is determined
, then make
Elements of A, p and VαThere is a one-to-one correspondence.In this wayαIs called the position vector of p starting from O
express.[1]
explain
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① The position vector of the particle in the selected coordinate system within the reference system is a directed line segment pointing from the origin of the coordinate system to the location of the particle, as shown in Figure 1.
② ForRectangular coordinate system, the position vector of the particle can be determined by x, y, z, and its size is
。The cosines of their directions are
, and
。
Figure 1 Position vector
Difference from displacement
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A potential vector describes the position of a moving particle in space at a certain time;The displacement describes the magnitude and direction of the position change of the moving particle in a certain time interval.Position vector corresponds to time;The displacement corresponds to the time interval.
Vector operation
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1. The addition of vectors A and B is defined as the sum of two vectors, and the new vector is usedA+Bexpress.UsedParallelogram ruleOr head to tail
Subtraction between A and B is defined as the difference between two vectorsA-Bexpress.Write asA-B=A+(-B）, press B to add with A in reverse direction.
Vector addition (subtraction) algorithm:
Exchange law:A+B=B+A；
Binding law:A+B-C=A+(B-C)=(A+B)-C。
2. Scalar and vectorAThe product of is defined as a new vector ƒA, which is ten times of A.
3. Two vectorsAandBThe scalar product of is defined as scalar, also called dot product.Its value is the cosine product of the angle α (0 ≤ α ≤ 180 °) between the module of two vectors and the two vectors.
characteristic:
(1) The dot product of two vectors is a scalar, and its positive and negative depend on whether α is an acute angle or an obtuse angle;
(2) The dot product obeys the commutative law;
（3）AAndBPerpendicular to each other|A||B|Cos α=0, and vice versa;
(4) Point product operation of A and B in rectangular coordinates: multiply each component of two vectors item by item.The dot product of the vector follows the distribution rate.
fourAandBThe vector product of is expressed asA×B, also called cross product.
characteristic:
(1) The cross product of two vectors is a vector;
(2) The cross product does not comply with the exchange rate. It should beA×B=-(B×A)；
（3）A、BWhen parallel (α=0 or 180 °),A×B=0, and vice versa ---- necessary and sufficient condition for two vectors to be parallel;
(4) The cross product of A itself is zero, that isA×A=0。
Relative position vector
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The relative position vector can represent the position relationship between any two points in space.R is a space vector with P 'as the starting point and P' as the ending point. Its modulus represents the distance between P 'and P', and its direction represents the orientation of P 'relative to P', so R is called the relative position vector of P 'relative to P'.
If the relative position vector of point P 'relative to point P is consideredR', thenR'The direction of is from point P to point P ', withR'=-R。
Any real physical field has its source, the so-called field source. For example, the static charge is the field source of the electrostatic field, the constant current is the field source of the constant magnetic field, and so on.The field source and the physical field it produces are always connected with the concept of space.In the future, we will study the relationship between the electromagnetic field and its source. The point where the field source is located and the point where the field quantity (such as electric field intensity vector and magnetic field intensity vector) needs to be determined need to be clearly distinguished in name and symbol.The point where the field source is located is called the source point for short, and is represented by the coordinates of the prime source point (x ', y', z ') or r';The point where the field quantity needs to be determined is referred to as the field point, which is represented by the coordinates (x, y, z) or r of the field point without prime.Therefore, R has the special meaning of the relative position vector of the field point relative to the source point.
The relative position vector of ordinary two points in space can be distinguished by adding double subscripts. For example, the relative position vector of P2 point to P1 point is marked as R12, and its direction is from P1 point to P2 point.
Relative coordinate function
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A class of functions related to the relative position vector, whose variable is the coordinate difference between the field point and the source point.Relative coordinate scalar function and relative coordinate vector function are respectively recorded as
