Addition & Scaling

Vector Addition

Definition:
The sum of any two vectors \( \color{vred}\overrightarrow{v} \) and \(\color{vblue} \overrightarrow{w} \) results in a third vector\( \color{vgreen} \overrightarrow{z} \). The components of \( \color{vgreen} \overrightarrow{z} \) are the sum of \( \color{vred}\overrightarrow{v} \) and \(\color{vblue} \overrightarrow{w} \) components.

\( \color{vred}\overrightarrow{v} + \color{vblue}\overrightarrow{w} \color{black} = \color{vgreen} \overrightarrow{z}\)

\( \color{vred} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} + \color{vblue} \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} = \color{vgreen} \begin{bmatrix} z_1 \\ z_2 \\ \vdots \\ z_n \end{bmatrix} \)

\( \color{vgreen}\overrightarrow{z} = \begin{bmatrix} {\color{vred}v_1} + {\color{vblue}w_1} \\ {\color{vred}v_2} + {\color{vblue}w_2} \\ \vdots \\ {\color{vred}v_n} + {\color{vblue}w_n} \end{bmatrix} \)

Examples:

\( \begin{bmatrix} 1 \\ 1 \end{bmatrix} + \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \)

\( \begin{bmatrix} -1 \\ 0 \end{bmatrix} + \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \)

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Addition as Translating Points

Vector addition can also be viewed as a process of translating points. To the right you can view the addition of two randomly generated vectors. This method of adding vectors is called the tip to tail method because you translate the tail of one vector along the length of the other until they are connected tip to tail.

Try it Yourself

You can use the interactive tool below to visualize adding two vectors.

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Change Dimension

\( {\color{vred}\overrightarrow{a}} + \color{vblue}\overrightarrow{b} \)

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Vector Scaling

Definition:
If \( \overrightarrow{v} \) is a vector and if \(k \) is a scalar then we define the scalar product of \( \overrightarrow{v} \) by \( k \) to be the vector whose length is \(|k| \) times the length of \( \overrightarrow{v} \) and whose direction is the same as that of \( \overrightarrow{v} \).

\( { \color{vblue} \overrightarrow{w} } = k\color{vred}\overrightarrow{v} \)

\( \color{vblue} \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} = \color{vred}k \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ z_n \end{bmatrix} \)

\( \color{vblue} \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} = \color{vred} \begin{bmatrix} kv_1 \\ kv_2 \\ \vdots \\ kv_n \end{bmatrix} \)

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Scaling as Stretching

When a vector is multiplied by a scalar it can be viewed as stretching the vector. Scaling a vector by a large positive number will elongate the vector while multiplying by a small fraction will shorten the vector.

Examples:

\( 2\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \end{bmatrix} \)

\( -1\begin{bmatrix} 3 \\ -1 \end{bmatrix} = \begin{bmatrix} -3 \\ 1 \end{bmatrix} \)

Try it Yourself

You can use the interactive tool below to visualize adding two vectors.

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Change Dimension

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Addition & Scaling

Vector Addition

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Addition as Translating Points

touch_appTry it Yourself

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Vector Scaling

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Scaling as Stretching

touch_appTry it Yourself

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

Try it Yourself

Try it Yourself