# Scalar-vector multiplication

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Geometric interpretation.

The **product of non-zero vector by the number**is a vector collinear with given (codirectional given if the number is positive, having the opposite direction if the number is negative) and its magnitude is equal to the module of the vector multiplied by the module of number.

Algebraic interpretation. The

**product of non-zero vector by the number**is a vector which coordinates are equal to the corresponding coordinates of the vector, multiplied by the number.## Scalar-vector multiplication - formulas

### The formula multiplying the vector by a number for plane problems

In the case of the plane problem the product of vector a = {a_{x} ; a_{y}} by the number k can be found using the following formula:

k · a = {k · a

_{x}; k · a_{y}}### The formula multiplying the vector by a number for spatial problems

In the case of the spatial problem the product of vector a = {a_{x} ; a_{y} ; a_{z}} by the number k can be found using the following formula:

k · a = {k · a

_{x}; k · a_{y}; k · a_{z}}### The formula multiplying the vector by a number for n dimensional space problems

In the case of the n dimensional space problem the product of vector a = {a_{1} ; a_{2}; ... ; a_{n}} by the number k can be found using the following formula:

k · a = {k · a

_{1}; k · a_{2}; ... ; k · a_{n}}## Properties of a vector multiplied by the number

If the vector b is equal to the product of a non-zero integer k and non-zero vector a, then:

- b || a - vectors b and a are parallel
- a↑↑b, if k > 0 - vectors b and a are codirectional, if the number k > 0
- a↑↓b, if k < 0 - vectors b and a is oppositely directed, if the number k < 0
- |b| = |k| · |a| - the magnitude of the vector b is equal to the module of the vector a multiplied by the module of number k

## Examples of tasks

### Examples of plane tasks

Example 1. Find the product of vector a = {1; 2} by 3.

**Solution:** 3 · a = {3 · 1; 3 · 2} = {3; 6}.

### Examples of spatial tasks

Example 2. Find the product of vector a = {1; 2; -5} by -2.

**Solution:** (-2) · a = {(-2) · 1; (-2) · 2; (-2) · (-5)} = {-2; -4; 10}.

Vectors
Vectors Definition. Main information
Component form of a vector with initial point and terminal point
Length of a vector
Direction cosines of a vector
Equal vectors
Orthogonal vectors
Collinear vectors
Coplanar vectors
Angle between two vectors
Vector projection
Addition and subtraction of vectors
Scalar-vector multiplication
Dot product of two vectors
Cross product of two vectors (vector product)
Scalar triple product (mixed product)
Linearly dependent and linearly independent vectors
Decomposition of the vector in the basis

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