Sum of Vectors

Two vector in 2-dimensional space:

\begin{aligned} \mathbf{a} &= (a_1, a_2), \text{ in }\mathbb{R}^2\\ \mathbf{b} &= (b_1, b_2), \text{ in }\mathbb{R}^2 \end{aligned}

The sum of two vectors is the vector resulting from the addition of the components of the original vectors:

\begin{aligned} \mathbf{c} &= \mathbf{a} + \mathbf{b}\\ &= (a_1 + b_1, a_2 + b_2) \end{aligned}

A sample:

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You can drag the two red dots to change the vectors $\mathbf{a}$ and $\mathbf{b}$ and see how this affects the result of the sum, that is vector $\mathbf{c}$

Difference of Vectors

Two vectors in 2-dimensional space:

\begin{aligned} \mathbf{a} &= (a_1, a_2), \text{ in }\mathbb{R}^2\\ \mathbf{b} &= (b_1, b_2), \text{ in }\mathbb{R}^2 \end{aligned}

The difference of two vectors is the vector resulting from the differences of the components of the original vectors:

\begin{aligned} \mathbf{c} &= \mathbf{a} - \mathbf{b}\\ &= (a_1 - b_1, a_2 - b_2) \end{aligned}

A sample:

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You can drag the two red dots to change the vectors $\mathbf{a}$ and $\mathbf{b}$ and see how this affects the result of the difference, that is vector $\mathbf{c}$

Scalar multiplication

A vector $a$ in 2-dimensional space and a scalar $\\lambda$ (a number):

\begin{aligned} \mathbf{a} &= (a_1, a_2), \text{ in }\mathbb{R}^2\\ \lambda \end{aligned}

A vector multiplied by a scalar is the vector resulting of the multiplication of each component of the original vector by the scalar:

\begin{aligned} \mathbf{c} &= \lambda \mathbf{a}\\ &= (\lambda a_1, \lambda a_2) \end{aligned}

A sample:

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You can drag the red dot to change the vector $\mathbf{a}$ and see how this affects the result of the scalar multiplication, that is vector $\mathbf{c}$

Change the value of scalar $\lambda$ by dragging the slider.