# Vector Addition

Solve problem in this form \[(a_1 + b_1)i + (a_2 + b_2)j+ (a_3 + b_3)k \]

## Vector Addition

A vector addition calculator computes the resultant vector by adding the corresponding components of two or more vectors, providing the magnitude and direction of the sum.

### Lesson Note: Vector Addition

#### Introduction to Vectors

A vector is a mathematical entity that has both magnitude (length) and direction. Vectors are often represented as arrows in a coordinate plane, where the length of the arrow indicates the magnitude and the direction of the arrow shows the direction of the vector.

**Notation**

Vectors are typically denoted by bold letters or letters with an arrow on top, such as a or \(\vec{a}\). In component form, a vector a in two dimensions can be written as:

\[

\vec{a} = \begin{pmatrix} a_x \\ a_y \end{pmatrix}

\]

where \( a_x \) and \( a_y \) are the components of the vector along the x-axis and y-axis, respectively.

#### Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. This can be done graphically or algebraically.

##### Graphical Method

**Tip-to-Tail Method**:- Place the tail of the second vector at the tip of the first vector.
- The resultant vector (sum) is drawn from the tail of the first vector to the tip of the second vector.

**Parallelogram Method**:- Place the two vectors so that they start from the same point.
- Complete the parallelogram formed by these vectors.
- The diagonal of the parallelogram from the common starting point is the resultant vector.

##### Algebraic Method

To add two vectors algebraically, simply add their corresponding components.

If \(\vec{a} = \begin{pmatrix} a_x \\ a_y \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} b_x \\ b_y \end{pmatrix}\), then the sum \(\vec{c} = \vec{a} + \vec{b}\) is:

\[

\vec{c} = \begin{pmatrix} a_x + b_x \\ a_y + b_y \end{pmatrix}

\]

#### Example

Let’s add two vectors \(\vec{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\).

Using the algebraic method:

\[

\vec{a} + \vec{b} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} + \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 + 1 \\ 4 + 2 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}

\]

The resultant vector \(\vec{c} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\) has components 4 in the x-direction and 6 in the y-direction.

Vector addition is an essential concept in physics and engineering, enabling the combination of forces, velocities, and other vector quantities. Understanding both graphical and algebraic methods of vector addition provides a comprehensive foundation for solving related problems in these fields.

#### Summary

– Vectors have both magnitude and direction.

– Vector addition can be performed graphically (tip-to-tail or parallelogram methods) or algebraically by adding corresponding components.

– Ensure vectors have the same dimensions when performing addition.