Unit Vectors and Vector Representation

Unit Vectors

• A "cap" notation (e.g., $\hat{a}$) indicates a unit vector, which has a magnitude of 1.

• A unit vector specifies a direction.

• In any given direction, there are infinite possible vectors, but only one unit vector.

• Unit vectors are used to define standard directions instead of using terms like east, west, north, and south.

Standard Unit Vectors

• Standard directions are defined as x, y, and z.

• The unit vector for the x-axis is denoted as $\hat{i}$.

• The unit vector for the y-axis is denoted as $\hat{j}$.

• The unit vector for the z-axis is denoted as $\hat{k}$.

• There are six directions: +x, -x, +y, -y, +z, and -z.

Mutually Perpendicular Directions

• The angles between x, y, and z axes are all 90 degrees.

• The standard unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ are mutually perpendicular.

Numerical Representation of Vectors

• Any vector $\vec{a}$ can be represented as the sum of three component vectors along the x, y, and z axes.

• $\vec{a} = a<em>x + a</em>y + a_z$.

• Any vector can be divided into multiple vectors (components).

Vector Components

• Every vector $\vec{c}$ can be divided into three vectors: one along the x-axis, one along the y-axis, and one along the z-axis.

• This standard method allows for easy comparison of vectors.

• The component along the x-axis is given by $a_x \hat{i}$.

• The component along the y-axis is given by $a_y \hat{j}$.

• The component along the z-axis is given by $a_z \hat{k}$.

• Therefore, a vector $\vec{a}$ can be written as $\vec{a} = a<em>x \hat{i} + a</em>y \hat{j} + a_z \hat{k}$.