Vector Addition

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

1. 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.

2. 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.