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} = ax + ay + 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} = ax \hat{i} + ay \hat{j} + a_z \hat{k} .