9Avectorintro


 * Introduction to Vectors**

**Vectors** are used to describe a variety of quantities that have direction as well as magnitude. (Eg displacement, velocity, accleration and force)

We represent a vector with a line segment that has an arrow on one end.
 * The length of the line segment represents the **magnitude** of the vector.
 * The angle and direction of the line segment shown by the arrow represents the **direction** of the vector.

Vectors are named
 * using capital letters at each end
 * or using a single lower case letter with a tilde (~) underneath.

Two vectors are **equal** if they have the same magnitude and direction.
 * {They can be shifted (translated) without changing equality}

Two vectors are **parallel** if they have the same direction but may have different magnitude.

If two vectors (**__a__** and **__b__**) are parallel, then they can be written as **__a__** = k**__b__**, where k is a constant.
 * {Note the underline should be a curved tilde (**~**) but limitations of the website software prevent me from showing that correctly}

A **unit vector** is a vector with a magnitude of one.

Vectors on Cartesian Axes {picture the z-axis coming straight up out of the page}
 * __i__** represents a unit vector in the direction of the x-axis.
 * __j__** represents a unit vector in the direction of the y-axis.
 * __k__** represents a unit vector in the direction of the z-axis.

These unit vectors are called the ** orthogonal unit vectors **. {orthogonal means mutually perpendicular ie they are all at right-angles to each other}

Using this notation, the dark blue vector shown on the right can be written as 2**__i__** + 4**__j__** {2 in the x direction and 4 in the y direction}

In three dimensions, this vector would be: 2**__i__** + 4**__j__** + 0**__k__**

Position Vectors

The **position vector for (a, b)** is a vector from the origin to a point (a, b) and would be written as a**__i__** + b**__j__**.

In above example (2**__i__** + 4**__j__**) would be the position vector for the point (2, 4) (shown in light blue).

Notice that the two vectors are ** equal ** (they have same magnitude and the same direction)

Matrix Notation Your calculator uses matrix notation for vectors: math \\ \text{The vector } 2\underline{i} + 4\underline{j} \qquad. \\ \text{can be written as a linear matrix: } \left[ \begin{matrix} 2 & 4 \end{matrix} \right] \qquad. math

math \\ \text{The 3D vector } 5\underline{i} - 2\underline{j} + \underline{k} \qquad .\\ \text{can be written as } \left[ \begin{matrix} 5 & -2 & 1 \end{matrix} \right] \qquad. math

See Introduction to Matrices

If using your calculator, don't forget to write your answer back in the normal __**i**__, **__j__** notation. .