__maths__>__Properties of Vector Arithmetics__>__Properties of Vector Multiplication by Scalar__### Applicable Properties and understanding them

In this page, learn about the applicable properties of multiplication of vector by scalar.

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For an arithmetic operation, usually, we learn the following properties

• closure,

• commutative,

• associative,

• distributive,

• Identity with respect to the operation

• Inverse with respect to the operation These properties are studied for addition and multiplication only.

When studying these properties, subtraction is considered to be inverse of addition, and division is considered to be inverse of multiplication.

If commutative property is to be applied for `a-b`, it has to be `a + (-b)` and commutative property of addition is applied.

If commutative property is to be applied for `a-:b`, it has to be `a xx (1/b)` and commutative property of multiplication is applied

If associative property is to be applied for `(a-b)-c`, it has to be `[a+(-b)]+ (-c)` and associative property of addition is applied.

If associative property is to be applied for `(a-:b)-:c`, it has to be `[a xx (1/b)] xx (1/c)` and associative property of multiplication is applied.

Apart from the 4 fundamental operations, for other arithmetic operations like modulus, these properties are not called using the standard names like closure property.

For example, take modulus of a number. Even though, it is established that modulus of a real number is a real number, it is not referred as closure property of modulus. The properties are exclusively studied for addition and multiplication.

The set of properties are studied for Addition and Multiplication.

Multiplication of vector by a scalar is defined between two different entities - a scalar and a vector. The applicable properties are given in the coming pages.

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