What is a negative vector

Vector algebra

For a precise description of many physical properties, absolute numbers with their units of measurement are not sufficient. For example, in the case of speed, magnetic or electric field strength, there is no direction in addition to the amount. Straight lines with the same length can be characterized by different orientations in the plane or in space. A vector is a directed quantity in which the measure and direction are combined. In symbolic form, vectors are represented by an arrow. The length of the arrow is the dimension or the amount and the arrowhead points in the direction in which the amount is pointing or has an effect. The following sections provide general and mathematical overviews in order to be able to work meaningfully with vectors.

The vector properties are followed by sections on component representation, addition and subtraction, vector multiplication, the amount of a vector, its normalization, the scalar product, information on the projection onto a vector, the vector product and the late product.

General properties of vectors

As vector symbols you can find the point identifier in capital letters under an arrow in the direction of the starting point to the end point or a lower case letter in Fraktur (German cursive) or a lower case letter under an arrow (Latin cursive). To represent the amount, the vector symbol is written in amount symbols.

A vector is clearly described by the position of its start and end point. The point distance is its amount. The unit of measurement belongs to the physical-technical vector description.

Generally speaking, the vector is a representative of the set of all vectors (arrows) with the same direction and length. These vectors form a vector class and are identical to one another if they can be brought into congruence by parallel shifting. They agree in amount (scalar) and direction, i.e. all components. The picture shows the vectors a and b as direction vectors, which are therefore the only representatives of their own vector class.

The amount, the length of the vectors shown in the picture, is a non-directional quantity and is called a scalar. It also includes the specification of the unit of measurement. Each of the two vectors can be shifted parallel to itself independently of the other without changing its properties. Vectors with these properties are called free or unbound vectors. If the shift is only allowed to take place along the direction of action, the vectors are volatile. Bound vectors run from the fixed starting point towards the end point.

It is an undirected quantity that is clearly described by a real number and unit of measurement.
Free vector - direction vector:
It is not tied to the coordinate zero point and is also known as an unbound vector. It is defined by its magnitude and direction and can be moved parallel to itself while maintaining these properties. Direction vectors can also be multiplied by a scalar. It connects two points in space or on the plane and is the shortest connection between two position vectors.
Bound vector:
It is a vector defined by its start and end point. It cannot be moved in parallel without changing its initial conditions. Starting from a fixed point, it is then clearly defined by its magnitude and direction.
Position vector:
It is a vector tied to the coordinate zero point and directed towards a defined end point. It cannot be shifted in parallel and cannot be multiplied by a scalar.
Zero vector:
It is a vector of length 0 with an indefinite direction.

Parallel vectors only have to agree in their direction. Antiparallel vectors have a 180 ° opposite sense of direction. Parallel and anti-parallel vectors are collinear vectors, because you can always place them on a common line by parallel shifting.

A vector with the same amount (and measure) and opposite direction is called an inverse or opposite vector. It is the anti-parallel vector to itself. Its components differ in their sign. The addition of the vector and the opposite vector results in a zero vector. The start and end points are identical, its amount is 0 and it has no sense of direction.

Vectors in the Cartesian coordinate system

Component representation

The numerical calculation with vectors is clearer if they are integrated into a coordinate system. The three-dimensional Cartesian coordinate system, where the three coordinate axes x, y and z form right angles to one another in pairs, has proven itself. If the vectors to be calculated lie in one plane, the z-axis is omitted and the plane, right-angled xy-axis system is used.

Unit or basis vectors are assigned to the three axes in the same direction. They are tied to the origin of coordinates as position vectors, with the amount 1 they are at right angles, orthogonal to each other. One uses for their symbols .

In the rest of the text, vectors are written using highlighted lower case letters without an arrow.

The picture shows the position vector a in the Cartesian coordinate system. It can be described as the sum of its vector components ax, ay, az, which are obtained by projecting the vector onto the coordinate axes. In the picture, these are also the position vectors that point from the origin of the coordinates in the respective axis directions.

An equivalent representation is the sum of the unit vectors multiplied by the respective scalar coordinates of the end point, which then also correspond to the vector components. Most commonly used is the notation as a column vector. In the running text it is easier to write it as a line vector a = (ax ay az).

If the vector is shifted parallel to itself, then its length, direction and sense of direction remain the same. The vector coordinates are unchangeable (invariant) during the parallel shift. In the case of a bound vector, the coordinates of the start and end point are fixed. The following applies to the component representation and the notation of the column vector of various vectors:

Addition and subtraction of vectors

If a sufficiently large force acts on a stationary mass, it will move in the direction of the force. If several forces act from the same point of application in different directions, then the movement and direction of the mass is different. The result can be determined with the help of vector calculation.

Vectors are directed quantities whose properties are retained when they are shifted parallel to themselves. This parallel shift allows the end and starting points of the vectors to be connected to one another. The picture shows two vectors that act in the plane from a point with the coordinates (2; 5). The resulting quantity is the sum vector which points from this starting point to the end point P (8; 5).

The two vectors R1R2 and B1B2 are in themselves free vectors. They can be written as row vectors r1 = (4 2) and b1 = (2 −2) or, as in the figure, as column vectors. If one shifts the vector b1 with its starting point B1 to the end point R2 of the vector r1 by parallel displacement, then the point coordinates change. Point B1 lies on R2, so it is additively shifted by 4 units horizontally and 2 units vertically. The coordinates of B2, which becomes point P (8; 5), change by the same units. The point coordinates of the shifted vector b2 are calculated by adding the vector coordinates r1 at which the shift is made to the coordinates of the start and end point.

The picture clearly shows that the vector r1 can also be shifted parallel to itself along the vector b1 with the same result. The end point of the vector r2 is also P (8; 5). The following applies to the sum vector: g = r1 + b2 = b1 + r2.

If the starting points of the vectors are in the origin of coordinates, then the addition result is identical for position vectors as well. The same also applies to vectors in the three-dimensional coordinate system. The result of the vectorial addition or subtraction is not equal to the sum or difference of the amounts (lengths) of the vectors, unless they are collinear.

Direction vector

The direction vector is determined by the coordinates of its start and end point. Position vectors can be given for these two points. The components of the direction vector are calculated from the components of the position vector to the end point, from which the components of the position vector to the starting point are to be subtracted.

Vector addition
Vectors are added by connecting the end point of a vector with the starting point of the vector to be added by parallel shifting. The sum vector is equal to the vector between the starting point of the first and the end point of the last vector.

Vector subtraction
Vectors are subtracted by adding the opposite vector of the vector to be subtracted.

Calculation rules for addition and subtraction

As in mathematics with numbers, the scalar quantities, the commutative and associative law also applies to the addition and subtraction of vectors. For the mathematical determination of the sum vector, the components of the individual vectors are added to or subtracted from one another.

Vector multiplication - multiplication by a scalar

If the same volatile vectors are strung together and added up, the result of the vector sum is equal to the multiplication of the number by the individual vector. The real number is a scalar quantity and is multiplied by the length of the vector. The direction of the vector is retained if the scalar is positive. The vectors run collinearly in parallel. If the scalar is negative, the direction is reversed and the angle between the volatile vectors is 180 °. The vectors run antiparallel to one another. Multiplication by zero gives the zero vector. The multiplication of a vector by a real scalar is usually referred to as S multiplication.

A vector is multiplied by a real scalar by multiplying its vector coordinates individually by the scalar.

Calculation rules for scalar multiplication

As with the multiplication of numbers, the associative and distributive law applies, where the scalars are elements of the real numbers. There is no commutative law because the multiplication of a number by a vector is not defined.

Amount of a vector

The projection of a vector maps its vector components onto each axis of the orthogonal Cartesian coordinate system. They can be written as multiples of the basis or unit vectors. The amount of the sum vector is determined using the vector components that are perpendicular to one another and the Pythagorean theorem. The solution uses the scalar multiplication of the vector components by the unit vectors, the squares of which have the scalar value 1. The result is a scalar value and has no unit of measure as a measure of its length.

The amount of a vector is calculated from the square root of the sum of the squares of its vector coordinates.

Normalization of a vector

The normalization is important for representations in the coordinate system. In the spatial Cartesian coordinate system, the unit or base vectors have the magnitude 1 and lie in the axis directions. Each vector that is not a zero vector can be assigned a unit vector running in its direction with the magnitude 1. The standardization of the vector gives its unit vector. An e indexed with the vector letter or the vector letter with a roof (circumflex) is written as a vector symbol or its identifier.

If an angle of intersection between two vectors can be calculated, the bisector can be determined using the normalized vectors. The sum vector of these unit vectors lies in the direction of the bisector. The picture shows the geometric proof of the statement for the unit vectors pointing in the direction of their not to scale vectors a and b. The unit vectors with the magnitude 1 span a rhombus with the sum vector w, which lies in the direction of the bisector.

Dot product - inner product

If two vectors enclose an angle, this can be calculated quite easily with the scalar product. In physics, for example, the change in energy along a path depends on the angle of attack of the force component along the path. The calculation is simplified with the scalar product. With the cosine law of plane trigonometry, the scalar product of two vectors is derived. Pythagorean theorem applies to right triangles and can be applied to any triangle if it is divided into two right triangles by a contour line.

Definition: The scalar product of two vectors is calculated from the product of the vector amounts multiplied by the cosine of the angle enclosed by the vectors.
Following sentence: The scalar product is calculated in the component representation as the sum of the multiplication of similar components.

If there is an acute angle between the two vectors, then the projection of one vector onto the other has the same sense of direction and the scalar product is a positive number. In the case of an obtuse angle, the projection is anti-parallel and the scalar product has a negative value. If both vectors form a right angle, then the scalar product is zero, since the cosine of 90 ° results in zero.

Calculation rules for the scalar product

The commutative law applies, where the result is independent of the order of the vectors. The distributive law applies where the same vectors can be excluded from sums and differences. The associative law applies with scalar quantities as an element of the real numbers.

The scalar product of a vector with itself is equal to the square of its absolute value. In the component representation it is calculated as the sum of all scalar vector components multiplied by itself: a · a = | a |2 = ax2 + ay2 + az2

If the scalar product of two vectors that are not zero vectors has the value 0, then they run orthogonally (at right angles) to one another.

The following example shows with two vectors in the plane that the calculation of the scalar product using the component representation and the cosine law have equal rights. Furthermore, the orthogonal position of two vectors is verified with the help of the scalar product.

Intersection angle of two vectors

The intersection angle of two vectors can be determined with the scalar product in the form of the cosine law. The equation is rearranged according to the cosine of the angle. The angle is calculated using the arc cosine. With cos (φ) = 0, φ = 90 ° is a right angle. With cos (φ)> 0 the result is an acute angle and with cos (φ) <0 an obtuse angle. At cos (φ) = ± 1, the vectors are collinearly aligned, parallel in the same direction at +1 and antiparallel at −1.

Projection of a vector onto a vector

Two vectors in a plane that are different from zero vectors can be shifted parallel to each other so that their starting points lie together. Each of the vectors can be mapped onto the other by perpendicular projection. The projection vector (red) lies in the direction of the vector onto which it was projected and only differs from it by a scalar value.

With the vertical projection, the component of the projected vector can be determined on a vector or on the axes of the Cartesian coordinate system.

Vector product - cross product

The vector product of two vectors is a new vector that is perpendicular to each of the two vectors. The three vectors are always arranged in a three-dimensional axis system. To distinguish it from the scalar product with the multiplication point as the arithmetic symbol, the vector product is written with a multiplication cross. The cross product is then read as a cross b and is also known as the outer product.

Two vectors that are not parallel in space and that are not zero vectors span a plane. The cross product of these vectors is perpendicular to this plane as a vector. Its amount is the measure of the area of ​​the parallelogram spanned by the two vectors. The three vectors form a legal system in sequence a, b, (a x b). If all three vectors begin at the same starting point and one would rotate the vector a to the vector b over the smaller angle, then the rotation in the direction (a x b) results in the sense of a right-hand screw.

The following figure shows the area calculation of a parallelogram spanned by the vectors a and b, not to scale. Starting from this, an equation for determining the vector product is to be derived that does not require any knowledge of the angle and only uses the vector coordinates. The basis is the Pythagorean theorem for trigonometric functions. In a secondary calculation, the expression under the root can be multiplied and simplified in Eq. (II), since only scalar vector components occur. Put the result together in such a way that the binomial theorem can be applied. With the equality in Eq. (III), Eq. (II) can be rewritten for the area.

The largest value of the vector product according to Eq. (I) results from the clamping angle of φ = 90 °. At an angle of φ = 0 °, both vectors are parallel to each other and the cross product has the value zero. This is also the case if at least one of the vectors is a zero vector.

If the scalar product is formed from the cross product with one of its two vectors, then zero must result, since the cross product is always perpendicular to the spanned area. The assertion is shown in the following image for a vector. With the scalar product for both vectors, the component representation of the cross product is then derived in a different way.

The vector p is the cross product of the vectors (a x b). With the two equations for the scalar product (a · p) and (b · p), two equations can be set up in the component notation.The three unknown components of the cross product appear in both equations. The system of equations can be solved if a fixed value is determined for an unknown.

In Eq. (V) everything is excluded, everything is reduced to a main denominator and multiplied, whereby the order of the scalar factors is free. Then by is excluded and Eq. (VI) is obtained. In this equation, find a suitable value for component px with which the components py and pz can be written in a simplified manner.

Vector product in the determinant representation

The scalar components of the cross product vector p = a x b agree with the differences determined above. If the result looks like the solution scheme for a determinant, there is no need to remember the correct indexing. If the vector components of spatially arranged vectors are written in their row form, the Sarrus rule can be applied to the resulting determinant.

If determinants are unknown, the following solution scheme can be helpful for the cross product of two vectors. The vector components are written in the familiar columnar form. You start at the top and think of the 1st line deleted. The diagonal values ​​of lines 2 and 3 from left to right are multiplied and the product of the diagonal values ​​from right to left is subtracted from this. In the next step, think of the 2nd line deleted. This time the difference between the products of the 3rd line and the 1st line is to be formed. The last pair of values ​​is calculated when the 3rd line is deleted from the diagonals of lines 1 and 2. In the products, the order of the scalar values ​​is free.

A similar, perhaps more easily noticeable solution scheme is shown in the following picture. The vector components written in column form are extended downwards by the two components above. Starting from the second line, the product with the following line is written diagonally from top left to right and the diagonal product is subtracted from bottom left to right. The scheme is retained for the following lines and added to the previous block.

Calculation rules for the vector product

Since the three vectors of the cross product form a right system, there is no commutative law. If you swap the two vectors in the cross product, the result vector points in the opposite direction. This can also be called the anti-commutative law. The distributive law and the associative law with real scalars are applicable.

The cross product of two vectors different from the zero vector is always zero if both vectors are collinear. For the three unit vectors of the three-dimensional right-angled axis system, there is a right-hand system according to the following scheme:

Late product

To calculate the volume of a three-dimensional body, information on the base area and height is necessary. This corresponds to three different directions and should therefore also be possible with vectors. The area can be determined with the vector or cross product. The height stands at a certain angle on this surface and points in the direction of the third dimension. If the three direction vectors are perpendicular to one another, they form the edges of a rectangle or, if the length is the same, a cube.

If the orientation is at an oblique angle, a parallelepiped is created made up of six parallelograms. A very well-known example is calcite. The calcium carbonate CaCO3 forms spatula crystals as large oblique-angled cuboids. In school physics experiments on light birefringence are shown on calcite crystals. According to the crystal form, this mixed product is also called a late product.

Two edges a and b of a spar crystal span the base area as vectors. The vector product is perpendicular to it as a surface vector. The crystal volume is determined by the third edge at an oblique angle to the surface. If this edge vector c is projected perpendicularly onto the surface vector, then the projection length is equal to the height of the spade. It can be calculated from the scalar product of the area vector (a x b) with the edge vector c and the angle included by both.

The spar volume is calculated as a mixed or scalar product from the vectors, the projection of which stands for the height with the area vector, the cross product of the two vectors that span the plane. The amount of the spatial product is equal to the volume measure of the body, which is spanned by the three space vectors. The value is positive if the three output vectors form a legal system in the specified order, otherwise negative. The third vector, here c, forms the inclusion angle φ with the normal vector of the surface. At an angle of 90 °, the volume is zero. The three vectors are coplanar and linearly dependent, since they lie in one plane.

The late product (c a b) of the three vectors is the scalar product of the vector c with (a x b).

Knowledge of the angle is not necessary to calculate the late product with the component representation of the vectors. The same applies to the calculation from the determinant notation using the Sarrus rule. In the determinant, the vector components can be arranged in the form of rows or columns.

Calculation rules for the late product

There is no commutative law. The vector product is always bracketed in the late product, since the expression (a · b) x c does not make any sense. The value of the late product remains the same if the three vectors are cyclically exchanged, but the arithmetic symbols remain in their place. If two vectors are swapped in the cross product, a sign change takes place. The value of the late product does not change when the arithmetic symbols are exchanged if the order of the vectors is retained.