How do you memorize the formula for two vector perpendicular parallel coordinates?

If vector 1 is (a,b).

Vector 2 is (c,d).

The vectors are perpendicular to each other.

then a c + b d = 0

If parallel, then a c = b d

Perpendicular is the point multiplied by 0, and each coordinate component corresponds to multiplication and addition. So the vertical formula is x1x2 + y1y2 + z1z2 = 0.

It's easy to remember. Let the two vector coordinates represent (x1, y1, z1) and (x2, y2, z2) (both are not zero vectors).

Perpendicular is the multiplication of points by 0, just remember the definition of point multiplication: each coordinate component corresponds to multiplication and addition. So the vertical formula is x1x2 + y1y2 + z1z2 = 0.

Parallel is easier to remember, that is, the corresponding coordinate components are proportional, x1:x2=y1:y2=z1:z2

1) The non-0 vectors a,b are parallel, i.e.: a b is sufficient and necessary for the existence of a real number ≠0 such that: a = b.

Let a=(x1,y1) b=(x2,y2) and a b, then there is 0 such that a= b, i.e.

x1,y1)= (x2,y2) -x1 x2=y1 y2= , so: x1y2=x2y1, i.e.: x1y2-x2y1=0;

2) Non-0 vectors a,b perpendicular, i.e.: a b: according to the formula of the product of vector quantities:

ab = |a| |b|cos(1) or.

ab = (x1x2+y1y2) (2)

1) is the angle between the vectors of a and b, when = 90° or = 2, ab = 0 and then by equation (2), we get: x1x2 + y1y2 = 0.

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)。

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.

It's easy to remember. Let the two vectors sit on the rock Wang mark, which are (x1, y1, z1) and (x2, y2, z2) (both are not zero vectors) Shichang.

Perpendicular is the multiplication of points by 0, just remember the definition of coarse return multiplication: each coordinate component corresponds to multiplication and addition. So the vertical formula is x1x2 + y1y2 + z1z2 = 0.

Parallel is easier to remember, that is, the corresponding coordinate components are proportional, x1:x2=y1:y2=z1:z2

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

Coordinates are represented as: a=(x1,y1), b=(x2,y2), a b if and only if x1y2-x2y1=0, a b if and only if x1x2+y1y2=0. Filial piety to God.

Since any set of parallel vectors can be moved to the same straight line, parallel vectors are also called collinear vectors.

Vectors that are equal are necessarily parallel, but vectors that are parallel are not necessarily equal. Just because two vectors are equal doesn't necessarily have to coincide. Just use the two vectors that are of equal length and in the same direction. "Same direction" implies the parallel of vectors.

In mathematics, a vector (also known as a Euclidean vector, geometric vector, 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 a vector is called a quantity (in physics, it is called a coincidence loss), and a quantity (or scalar) is only a magnitude and has no direction.

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'...