# Basic Properties of Vectors

We will now look at some important definitions and properties of vectors in Euclidean n-space. All proofs given are rather straightforward from the definition of the operations we are about to look at, and most explanation will be omitted.

Definition: Two vectors $\vec{u}$ and $\vec{v}$ are said to be Equal if and only if each of their respective components are equal, that is $u_{i} = v_{i}$ for every vector component $i$. |

For example, the following vectors $\vec{u} = (1, 3)$ and $\vec{v} = (1, 3)$ are equal since $u_1 = v_1 = 1$ and $u_2 = v_2 = 3$.

Definition: If $\vec{u}, \vec{v} \in \mathbb{R}^n$ are vectors, then Vector addition is defined by adding corresponding components to each other, that is $\vec{u} + \vec{v} = (u_1, u_2, …, u_n) + (v_1, v_2, …, v_n) = (u_1 + v_1, u_2 + v_2, …, u_n + v_n)$. Furthermore, vector subtraction is defined by $\vec{u} – \vec{v} = (u_1, u_2, …, u_n) – (v_1, v_2, …, v_n) = (u_1 – v_1, u_2 – v_2, …, u_n – v_n)$. |

We will now look at the important commutativity property of vector addition.

Theorem 1 (Commutativity of Vector Addition): If $\vec{u}, \vec{v} \in \mathbb{R}^n$, then $\vec{u} + \vec{v} = \vec{v} + \vec{u}$. |

**Proof:**Let $\vec{u}, \vec{v} \in \mathbb{R}^n$ The addition of vectors is defined such that each respective component of vector the first vector will be added to the second vector, that is:

(1)

For example, if we wanted to find the sum $\vec{u} + \vec{v}$ given that $\vec{u} = (1, 2)$ and $\vec{v} = (4, 2)$, then:

(2)

Furthermore, if we instead calculate $\vec{v} + \vec{u}$, we get the same result:

(3)

Definition: The Zero Vector in Euclidean n-space denoted $\vec{0} = \underbrace{(0, 0, …, 0)}_{\mathrm{n-times}}$ has all of its components as zeroes. |

Theorem 2 (Zero Vector Addition Property): If $\vec{u}, \vec{0} \in \mathbb{R}^n$, then $\vec{u} + \vec{0} = \vec{0} + \vec{u} = \vec{u}$. |

**Proof:**From Theorem 1, we have that $\vec{u} + \vec{0} = \vec{0} + \vec{u}$ already, so we only need to show that $\vec{u} + \vec{0} = \vec{u}$.

(4)

Theorem 3 (Associativity of Vector Addition): If $\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^n$, then it follows that $\vec{u} + ( \vec{v} + \vec{w} ) = ( \vec{u} + \vec{v} ) + \vec{w}$. |

**Proof:**

(5)

Recall that a scalar is defined as a quantity that has only magnitude and no direction. We will now look at our next definition.

Definition: If $\vec{u} \in \mathbb{R}^n$, and $k \in \mathbb{R}$, then $k\vec{u}$ is defined to be a Scalar Multiple of the Vector $\vec{u}$ that has a length that is $k$ times that of $\vec{u}$, and $k\vec{u} = k(u_1, u_2, …, u_n) = (ku_1, ku_2, …, ku_n)$. If $k , the direction of the vector changes to be in the position opposite to $\vec{u}$. |

Theorem 4: If $\vec{u} \in \mathbb{R}^n$ and $k, l$ are scalars such that $k, l \in \mathbb{R}$, then $k(l\vec{u}) = (kl)\vec{u}$. |

**Proof:**

(6)

Theorem 5: If $\vec{u} \in \mathbb{R}^n$ and $k, l$ are scalars such that $k, l \in \mathbb{R}$, then $(k + l)\vec{u} = k\vec{u} + l\vec{u}$. |

**Proof:**

(7)

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