1. Meaning of Vector Addition
Vector addition combines two or more vectors into a single resultant vector. The idea is to place vectors head-to-tail and see where the final arrow ends up.
The resultant vector shows the overall effect when both vectors act together.
2. Triangle Rule of Vector Addition
The triangle rule says: place the tail of the second vector at the head of the first vector. The vector from the tail of the first to the head of the second is the sum.
This creates a triangle with the resultant as its third side.
2.1. Example
If \(\vec{u}\) and \(\vec{v}\) are two vectors, then:
\[ \vec{u} + \vec{v} = \overrightarrow{\text{start of } u \;\to\; \text{end of } v} \]
For \(\vec{u} = \langle 2, 1 \rangle\) and \(\vec{v} = \langle 3, 2 \rangle\), you can place them head-to-tail to get the final direction easily.
3. Parallelogram Rule of Vector Addition
In the parallelogram rule, both vectors start from the same point. Complete the parallelogram, and the diagonal from the common starting point gives the sum.
This method is especially useful in diagrams or geometric problems.
3.1. Example
Draw both \(\vec{u}\) and \(\vec{v}\) from a common origin. Complete the parallelogram. The diagonal is:
\[ \vec{u} + \vec{v} \]
4. Component-wise Addition
Vector addition is simplest when done through components. Add x-components together, y-components together, and (if in 3D) z-components together.
If:
\[ \vec{u} = \langle a_1, b_1 \rangle, \quad \vec{v} = \langle a_2, b_2 \rangle \]
Then:
\[ \vec{u} + \vec{v} = \langle a_1 + a_2,\; b_1 + b_2 \rangle \]
In 3D:
\[ \langle a_1, b_1, c_1 \rangle + \langle a_2, b_2, c_2 \rangle = \langle a_1+a_2, b_1+b_2, c_1+c_2 \rangle \]
4.1. Example
For \(\vec{u} = \langle 4, -1, 2 \rangle\) and \(\vec{v} = \langle 1, 3, -2 \rangle\):
\[ \vec{u} + \vec{v} = \langle 4+1,\; -1+3,\; 2+(-2) \rangle = \langle 5, 2, 0 \rangle \]
5. Resultant Vector and Magnitude
Once two vectors are added, you can find the magnitude of the resultant using the distance formula.
If the resultant is \(\vec{R} = \langle x, y \rangle\), then:
\[ |\vec{R}| = \sqrt{x^2 + y^2} \]
5.1. Example
If \(\vec{u} = \langle 2, 3 \rangle\) and \(\vec{v} = \langle 1, 1 \rangle\):
\[ \vec{u} + \vec{v} = \langle 3, 4 \rangle \]
Magnitude:
\[ |\vec{u} + \vec{v}| = 5 \]
6. Important Notes
- Vector addition is commutative: \( \vec{u} + \vec{v} = \vec{v} + \vec{u} \)
- It is also associative: \( (\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}) \)
- The zero vector acts as the identity element: \( \vec{u} + \vec{0} = \vec{u} \)