Adding vectors results in a new Resultant Vector. Geometric and Algebraic vectors can be added using different techniques.
Algebraic Vectors can be added by adding the respective components along each axis.
As an example. we can add the following algebraic vectors to get the resultant vector:
\( \vec{a} = \left<3,4\right>, \; \vec{b} = \left<-4,10\right> \)
\( \vec{a} + \vec{b} = \left<3 + -4, 4 + 10\right> = \left<-1, 14\right> \)
Recall the vector addition rules:
| Commutative Law of Addition | \(\vec{u} + \vec{v} = \vec{v} + \vec{u}\) |
| Associative Law of Addition | \( (\vec{m} + \vec{n}) + \vec{o} = \vec{m} + (\vec{n} + \vec{o})\) |
We can determine \(2\vec{d} + 3\vec{e}\) as such:
\(2 \vec{d} + 3 \vec{e} = 2 \left<3,6\right> + 3 \left<-4,-5\right>\)
\(= \left<2 \times 3, 2 \times 6\right> + \left<3(-4), 3(-5)\right>\)
\(= \left<6,12\right> + \left<-12,-15\right>\)
\(= \left<6 + (-12),12 + (-15)\right>\)
\(= \left<-6,-3\right>\)
Therefore, we can determine that \(2\vec{d} + 3\vec{e}\) is \(\boldsymbol{\left<-6,-3\right>}\).
Like adding Algebraic vectors, subtract the respective components along each axis:
\( \vec{a} = \left<3,4\right>, \; \vec{b} = \left<-4,10\right> \)
\( \vec{a} - \vec{b} = \left<3 - (-4), 4 - 10\right> = \left<7, -6\right> \)
You can think of subtracting vectors as adding the Opposite vector:
\( \vec{a} - \vec{b} = \vec{a} + (-\vec{b}) \)
\( = \left<3,4\right> + (-\left<-4,10\right>) \)
\( = \left<3,4\right> + \left<4,-10\right> \)
\( = \left<3+4,4-10\right>\)
\( = \left<7,-6\right>\)
Enter in two vectors below. Then, Click the + button to toggle between addition and subtraction.