Given two vectors u and v in Rn $$u=(a_1,a_2,...,a_n) \\ u=(b_1,b_2,...,b_n)$$ the sum u + v p is a vector in Rn obtained by adding the corresponding components of u and v $$u+v=(a_1+b_1,a_2+b_2,...,a_n+b_n)$$

The sum of vectors with a different number of components is not defined.

Example

Given two row vectors in R3

$$u=(3,4,1)$$

$$v=(1,-1,1)$$

The sum of the vectors u + v is another vector in R3

$$u+v=(3+1,4+(-1),1+1) = (4,3,2)$$

Example

Given two column vectors in R3

$$u= \begin{pmatrix} 3 \\ 4 \\ 1 \end{pmatrix}$$

$$v=\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}$$

The sum of the vectors u + v is another vector in R3

$$u+v=\begin{pmatrix} 3 \\ 4 \\ 1 \end{pmatrix} + \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3+1 \\ 4+(-1) \\ 1+1 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \\ 2 \end{pmatrix}$$

## Theorems

• Vector addition satisfies the associative law for addition. Given three vectors u, v, w $$(u+v)+w=u+(v+w)$$
• Vector addition satisfies the commutative law for addition. Given two vectors u, v $$u+v=v+u$$
• Let v be a vector and o be the null vector in Rn, the addition v + o is equal to the vector v. $$v+o=v$$
• Let v be a vector and -v be the negative vector of v in Rn, the addition v + (-v) is equal to the null vector o. $$v+(-v)=o$$

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Vectors

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