Vector parallel coordinate formula, what is vector parallel formula

Two vectors a and b are parallel: a = b (b is not a zero vector); The two vectors are perpendicular: the quantity product is 0, i.e. a b=0.

Coordinate representation: a=(x1,y1), b=(x2,y2).

a b if and only if x1y2-x2y1=0

a b if and only if x1x2 + y1y2 = 0

In the Cartesian coordinate system, we take two unit vectors that are the same as the x-axis and y-axis.

i, j as the base. Make any vector a, by the plane vector fundamental theorem.

It can be seen that there is only one pair of real numbers x and y, such that a=xi+yj, and we call (x,y) the (rectangular) coordinates of the vector a, denoted as: a=(x,y).

where x is called a's coordinate on the x-axis, y is called a's coordinate on the y-axis, and the above equation is called the coordinate representation of the vector. in a planar Cartesian coordinate system.

, each plane vector can be uniquely represented by a pair of real numbers.

Two vectors parallel means that they are in the same direction or opposite. You can use the coordinate formula to determine if two vectors are parallel.

Suppose there are two vectors a = (a1, a2, a3) and b = (b1, b2, b3).

If a and b are parallel, the following formula can be used:

a1/b1 = a2/b2 = a3/b3

If both coordinate pairs hold in the above equation, i.e., a1 b1 = a2 b2 = a3 b3, you can determine that vectors a and b are parallel. Note that this formula requires that every coordinate of b be non-zero, otherwise it will result in a division error.

In addition, you can use the fork product formula of the vector to determine whether two vectors are parallel. If the cross product of two vectors results in zero vectors, they are parallel.

It is important to note that these formulas only work with vectors in three-dimensional space. In two-dimensional space, a similar method can be used to tell if the vectors are parallel, but the formula form will be different.

When using a coordinate formula or a cross-product formula to determine whether vectors are parallel, it is important to keep in mind numerical precision and rounding error to ensure accuracy.

Two vectors parallel means that they are in the same or opposite directions, and the coordinate representation of the vector can be used to determine whether the two vectors are parallel or not. Suppose there are two vectors:

Vector a:a = (a1, a2, a3).

Vector b:b = (b1, b2, b3).

The condition for two vectors to be parallel is that their coordinates are proportionally equal. That is, if there is a non-zero constant k, such that:

a1/b1 = a2/b2 = a3/b3 = k

This condition indicates that the proportions of the corresponding coordinates of vector a and vector b are equal. Note that if k = 0, then vectors a and b are collinear, but not necessarily parallel.

For example, if two vectors a = (2, 4, 6) and b = (1, 2, 3), we can calculate their coordinate scale:

a1/b1 = 2/1 = 2

a2/b2 = 4/2 = 2

a3/b3 = 6/3 = 2

Since all three scales are equal to 2, vector a and vector b are parallel.

It is important to note that this method only works with vectors in 3D space. In the case of higher dimensions, the condition of the coordinate scale is expanded accordingly.

There are two coordinates (x1, y1), (x, 2y2), and if parallel, then x1 x2 = y1 y2

Addition of vectors: ab+bc=ac, let a=(x,y) b=(x',y') then a+b=(x+x',y+y'The addition of vectors satisfies the law of parallelograms and triangles. Properties of Vector Addition:

Commutative property: a+b=b+aAssociative property: (a+b)+c=a+(b+c)a+0=0+a=a2, vector subtraction ab-ac=cba-b=(x-x',y-y'...

a×b=xn-ym=0

The vector is perpendicular, parallel to the equation:

If a and b are two vectors: a (x,y)b (m,n);

then the sufficient and necessary conditions of a b.

is a·b=0, i.e. (xm+yn)=0;

The formula for vector parallelism is: a b a b=xn-ym=0;

Vector introduction. The term "vector" comes from bonded cleft mechanics, directed line segments in analytic geometry. The first to use directed line segments to represent vectors was the great British scientist Isaac Newton.

From the history of the development of mathematics.

It was not until the late 19th and early 20th centuries that people associated the nature of space with vector operations, making vectors a set of numbers with excellent computational properties.

For vectors to enter mathematics and be developed, we should first start with the geometric representation of complex numbers. Late 18th century, Norway.

For the first time, the surveyor Wiesel used the points on the coordinate plane to represent the complex number a+bi (a, b are rational numbers and are equal to 0 at different times), and used the complex number operation with the meaning of several scum to define the operation of the vector.

Vector parallel formula coordinate formula: a= b, where b is not a zero vector. Coordinates are represented as: a=(x1,y1), b=(x2,y2), a b if and only if x1y2-x2y1=0.

In mathematics, vectors (also known as Euclid.

Vectors, geometric vectors, vectors), refers to quantities with magnitude and direction. It can be visualized as a line segment with an arrow. The arrow points to:

represents the direction of the vector; Line Segment Length: Represents the size of the vector. Quantities that correspond to vectors only in size, and quantities that have no direction are called quantities (scalars in physics).

If e1 and e2 are two non-collinear non-zero vectors in the same plane, then for any vector a in the plane of the flat search, there is only one and only one pair of real numbers, such that a = e1 + e2.

Given the space three vectors a, b, silver c, vector product of vectors a and b.

a b, and then with vector c as a quantity product (a b)·c, the resulting number is called the mixed product of the three vectors a, b, and c, which is denoted as (a, b, c) or (abc), that is, (abc) = (a, b, c) = (a b)c.

Mixed products have the following properties:

1. The absolute value of the mixed product of the three non-co-faceted quantities a, b, and c.

It is equal to a parallelepiped with a, b, and c as edges.

and when a, b, and c constitute the right-hand system.

when the mixed product is positive; When a, b, and c form the left-handed system, the mixed product is negative, i.e., (abc) = v (when a, b, c form the right-handed system =1;).When a, b, and c form a left-handed tie =-1).

2. The inference of the properties of the previous article: the sufficient and necessary conditions of the coplanarity of the three vectors a, b, and c.

Yes (abc) = 0.

3、(abc) =bca) =cab) =bac) =cba) =acb)。

Spatial vectorsThe coordinate formula of the flat head divination line formula: d=|ax0+by0+c|/√a^2+b^2。Quantities in space that have magnitude and direction are called space vectors.

The magnitude of a vector is called the length or modulus of the vector. Specify: A vector of length 0 is called a zero vector.

Write as 0. <>

Parallel judgment method of space vector:

Let the coordinates of one vector be (x,y,z) and the coordinates of the other vector be (a,b,c). If (x a) = (y b) = (z c) = constant, then the two vectors are parallel, and if ax+by+cz=0, then the two vectors are perpendicular.

If a=(x,y), b=(x'，y'If a b = 0 (a and b of the order of magnitude.)

Potato cherry is xx'+yy'=0, then a b. If a b = 0, then vector a is parallel to vector b; a=b, and a is also parallel to b. The hand spike.

The parallel formula is if a, b are two vectors: a=(x,y)b=(m,n); then a bSufficient conditionsis a·b=0, i.e. (xm+yn)=0;

The sufficient and necessary conditions of the vector a=(x1,y1), vector b=(x2,y2), x1y2-x2y1= are a·b=0, and the attack circle is (x1x2+y1y2)=0.

In mathematics, the vector (base remainder is also known as Euclid.

Vector, geometric vector, vector), refers to a quantity with magnitude and directionIf a = (x,y), b = (m,n), then a b a b b=xn-ym=0".

Parallel vectors. Non-zero vectors with the same or opposite direction are called parallel (or collinear) vectors Vectors a, b are parallel (collinear) and denoted a b. The length of the zero vector is zero, which is a vector that coincides with the start and end points, and its direction is uncertain.

We stipulate that the zero vector is parallel to either vector. A set of vectors parallel to the same line is a collinear vector.

The parallel formula of two coordinate vectors is x1y2=x2y1, where x1y1 is a coordinate point, x2y2 is a coordinate point, and coordinates refer to an ordered number or a set of numbers that can determine the position of a point on a plane or in space.

Parallel vectors. Also known as collinear vectors.

Refers to non-raffle zero vectors in the same or opposite direction, zero vectors are parallel to any source vector, vectors refer to quantities that have both magnitude and direction, and.

The zero liquid mill vector refers to a vector with a length of 0.

The coordinate cross multiplication corresponding to the parallel plane vectors is equal, i.e., x1y2=x2y, and the inner product is 0 in the vertical direction.

Zero vectors with the same or opposite direction are called parallel (or collinear) vectors. The vectors a and b are parallel (collinear) and are denoted as a b. The length of the zero vector is zero, i.e., the vector whose start and end points coincide and the direction is uncertain.

We stipulate that the zero vector is parallel to any vector. A set of vectors parallel to the same line is a collinear vector. The sufficient and necessary condition of a b is a·b=0, i.e., (x1x2+y1y2)=0.

Definitions

From the fundamental theorem of plane vectors, it can be seen that there is only one pair of real numbers (x,y) such that a=xi+yj, so the pair of real numbers (x,y) is called the coordinates of the vector a, and is denoted as a=(x,y). This is the coordinate representation of vector a. where (x,y) is the coordinates of the point.

The vector a is called the position vector of the point p.

Given two vector spaces v and w in the same f field, set up a linear transformation or "linear map" from v to w, and these v to w mappings have in common that they maintain a sum and a scalar quotient.

This set contains all linear images from v to w, described in l(v,w), and is also a vector space in an f-field. When v and w are determined, the linear mapping can be expressed in terms of matrices.

Two vectors a and b are parallel: a = b

b is not a zero vector); The two vectors are perpendicular: the quantity product is 0, i.e. a b=0 coordinates depresent: a=(x1,y1),b=(x2,y2)a b if and only if x1y2-x2y1=0

a b if and only if x1x2 + y1y2 = 0

The vector is perpendicular, parallel to the equation:

If a and b are two vectors: a (x,y)b (m,n);

Then the sufficient and necessary condition of a b is a·b=0, i.e., (xm+yn)=0;

The formula for vector parallelism is: a b a b=xn-ym=0;

In mathematics, a vector refers to a quantity with magnitude and direction. It can be visualized as a line segment with an arrow. The arrow points to:

represents the direction of the vector; Line Segment Length: Represents the size of the vector. The quantity corresponding to the vector is called the quantity (called the scalar in physics), and the quantity (or scalar) is only a magnitude and has no direction;

1. Vector vertical formula.

Vector a=(a1,a2), vector b=(b1,b2)a b: a1 b1=a2 b2 or a1b1=a2b2 or a= b ( is a constant).

a vertical b: a1b1 + a2b2 = 0

2. Vector parallel formula.

Vector a=(x1,y1), vector b=(x2,y2)x1y2-x2y1=0

The sufficient and necessary condition of a b is a·b=0, i.e., (x1x2+y1y2)=0.

a, b are two vectors.

a=（a1，a2） b=（b1，b2）

a b: a1 b1=a2 b2 or a1b1=a2b2 or a= b, is a constant.

a vertical b: a1b1 + a2b2 = 0

There are two vectors a and b, and the sufficient and necessary conditions for a b are a·b=0, i.e., (x1x2+y1y2)=0.