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Scalar-vector multiplication

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 = {ax ; ay} by the number k can be found using the following formula:

k · a = {k · ax; k · ay}

The formula multiplying the vector by a number for spatial problems

In the case of the spatial problem the product of vector a = {ax ; ay ; az} by the number k can be found using the following formula:

k · a = {k · ax ; k · ay ; k · az}

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 = {a1 ; a2; ... ; an} by the number k can be found using the following formula:

k · a = {k · a1; k · a2; ... ; k · an}

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}.

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