Properties of vectors

A vector is a quantity that has a magnitudeLength or strength of a vector. and a direction. In various areas of physics and mathematics, vectors are used to understand the behavior of directional quantities in two and three dimensional spaces. Moreover, they help us determine position and change in position of points.

Let's represent a vector $v$, written as $\vec{v}$, graphically now.

In this Answer, we will discuss the various properties of a vector.

1) Commutative property of vector addition

When two vectors are added to each other, changing their order keeps the result constant. Suppose we have two vectors $\vec{v}1$ and $\vec{v}2$, we can represent their commutative property of vector addition as:

Graphically, this can be presented as:

Example

Suppose we have two vectors $\vec{v1}$ and $\vec{v2}$, such that:
$\vec{v1} = (2,3)$

$\vec{v2} = (4,5)$

• Calculating $\vec{v1} +\vec{v2}$:

• Calculating $\vec{v2} +\vec{v1}$:

We conclude that the commutative property of vector addition holds.

2) Associative property of vector addition

When three vectors are added to each other, it doesn't matter which two vectors are added first. This means changing their order keeps the result constant. Suppose we have three vectors $\vec{v1}$, $\vec{v2}$ and $\vec{v3}$, we can represent their associative property of vector addition as:

Graphically, this can be presented as:

Example

Suppose we have three vectors $\vec{v1}$ , $\vec{v2}$, and $\vec{v3}$, such that:
$\vec{v1} = (2,3)$

$\vec{v2} = (4,5)$

$\vec{v3}=(1,2)$

• Calculating $\vec{v1} + (\vec{v2}+\vec{v3})$:

• Calculating $(\vec{v1} + \vec{v2})+\vec{v3}$:

We conclude that the associative property of vector addition holds.

3) Additive identity

A special vector is defined so that when a vector is added to it, the result is the vector itself. This special vector is a null vectorA vector with a magnitude of zero in all directions.. Suppose we have a vector $\vec{v1}$, we can represent its additive identity as:

Here $\vec{O}$ represents a null vector.

Example

Suppose we have a vector $\vec{v1}$and a null vector $\vec{O}$ such that:
$\vec{v1} = (2,3)$

$\vec{O}=(0,0)$

• Calculating $\vec{v1} +\vec{O}$:

We conclude that additive identity of a vector holds.

4) Additive inverse

When a vector is added to its negative vectorA vector with same magnitude but opposite direction., the resultant is a null vector. Suppose we have a vector $\vec{v1}$ and its negative vector $-\vec{v1}$, we can represent its additive inverse as:

Here $\vec{O}$ represents a null vector.

We can represent a vector and its negative vector as:

Example

Suppose we have a vector $\vec{v1}$, such that:
$\vec{v1} = (2,3)$

$-\vec{v1}=(-2,-3)$

• Calculating $\vec{v1} + (-\vec{v1})$:

We conclude that additive inverse property of a vector holds.

5) Multiplicative identity

If a vector is multiplied by a scalar $1$, the result is the vector itself. Suppose we have a vector $\vec{v1}$, we can represent its multiplicative identity as:

Here, $1$ is a scalar.

Example

Suppose we have a vector $\vec{v1}$, such that:
$\vec{v1} = (2,3)$

• Calculating $\vec{v1}× 1$:

We conclude that the multiplicative identity property of a vector holds.

6) Associative property of scalar multiplication

When we multiply a vector by a scalar and then multiply the result by another scalar, the order in which both the scalars are multiplied doesn't affect the result. Suppose we have a vector$\vec{v1}$, we can represent its associative property of scalar multiplication as:

Here, $a$ and $b$ are scalars.

Example

Suppose we have a vector $\vec{v1}$, such that:
$\vec{v1} = (2,3)$

$a = 5$

$b= 4$

• Calculating $a ×(b×\vec{v1})$:

• Calculating $( a×b) × \vec{v1}$:

We conclude that the associative property of scalar multiplication holds.

7) Distributive property over vector addition

When we multiply a scalar by the sum of two vectors, it's the same as multiplying each vector by that scalar separately and then adding the results together. Suppose we have two vectors $\vec{v1}$and $\vec{v2}$, its distributive property over vector addition is defined as:

Here, $a$ is a scalar.

Example

Suppose we have vectors $\vec{v}1$and $\vec{v}2$, such that:
$\vec{v1} = (2,3)$

$\vec{v2} = (4,5)$

$a = 5$

• Calculating $a ×(\vec{v1}+\vec{v2})$:

• Calculating $a \vec{v1} +a \vec{v2}$:

We conclude that distributive property over vector addition holds.

8) Distributive property over scalar addition

When we multiply a vector by the sum of two scalars, it's the same as multiplying each scalar by that vector separately and then adding the results together. Suppose we have a vector$\vec{v1}$, its distributive property over scalar addition is defined as:

Here, $a$ and $b$ are scalars.

Example

Suppose we have a vector $\vec{v1}$, such that:

$\vec{v1} = (2,3)$

$a = 5$

$b=4$

• Calculating $\vec{v1}×(a+b)$:

• Calculating $a \vec{v1} +b \vec{v1}$:

We conclude that distributive property over scalar addition holds.

Conclusion

Understanding the properties of vectors is crucial to understanding vectors' behaviors in two and three-dimensional spaces. They help us solve complex problems and make accurate predictions about the behavior of directional quantities.

Note: If you want to further study about the dot and cross product of two vectors you can refer to the following Answers: dot product and cross product.