
Adding Vectors in Linear Algebra
Today, we're addressing the topic of vector addition within the framework of linear algebra, taking a direct, example-based approach.
Suppose we are presented with two vectors $$ \vec{v_1} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} \\ \vec{v_2} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} $$ the sum of these vectors is a new vector whose components are the aggregate of the corresponding components of v and w, resulting in $$ \vec{v_1} + \vec{v_2} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1+x_2 \\ y_1+y_2 \end{pmatrix} $$
Consider, for example, two vectors,
$$ \vec{u} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \\ \vec{v} = \begin{pmatrix} 5 \\ 1 \end{pmatrix} $$
These vectors appear as such within the Cartesian plane.
To determine their sum, we need to aggregate the respective components,
$$ \vec{u} + \vec{v} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 5 \\ 1 \end{pmatrix} = \begin{pmatrix} 2+5 \\ 3+1 \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}$$
Consequently, the resulting vector is
$$ \vec{u} + \vec{v} = \begin{pmatrix} 7 \\ 4 \end{pmatrix} $$
The resulting vector, when represented in Cartesian coordinates, appears as follows:
If the vectors comprise two components, their sum can also be geometrically visualized through the parallelogram method or the tip-to-tail method.
It's essential to note that the length (magnitude or norm) of the resultant vector is always less than or equal to the sum of the magnitudes of the input vectors. This characteristic, known as the triangle inequality, is expressed as: $$ | \vec{v_1} + \vec{v_1} | \le |\vec{v_1}|+|\vec{v_2}| $$ The magnitude of the resultant vector equals the sum of the input vector magnitudes when the vectors are co-directional, or at least one of them is a zero vector.
But what if there are more than two vectors?
The calculation method remains consistent.
Suppose you are presented with three vectors,
$$ \vec{u} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \\ \vec{v} = \begin{pmatrix} 5 \\ 1 \end{pmatrix} \\ \vec{w} = \begin{pmatrix} -1 \\ 1 \end{pmatrix} $$
The resultant vector is then a new vector derived from the algebraic sum of the respective vector components,
$$ \vec{u} + \vec{v} + \vec{w} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 5 \\ 1 \end{pmatrix} + \begin{pmatrix} -1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2+5+(-1) \\ 3+1+1 \end{pmatrix} = \begin{pmatrix} 6 \\ 5 \end{pmatrix} $$
And what about vectors comprising more than two components?
The principle remains unchanged.
Suppose you have two vectors, each containing three components,
$$ \vec{u} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} \\ \vec{v} = \begin{pmatrix} 5 \\ 1 \\ 4 \end{pmatrix} $$
Their sum would be
$$ \vec{u} + \vec{v} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 5 \\ 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2+5 \\ 3+1 \\ 1+4 \end{pmatrix} = \begin{pmatrix} 7 \\ 4 \\ 5 \end{pmatrix} $$
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