Transforming from one coordinate system to another How can we relate r or q to x and y? x (x, y) Pythagorean Theorem r y y q Converts from Cartesian to polar coordinates. x (0, 0) x r is standard notation for a length that can contain x, y and z coordinates. You must know how to use these trigonometric functions!! Remember that positive angles are measured counterclockwise from 0o (usually from the positive x- axis). These definitions are based on the angle shown in the diagram. If you use a different angle you may have to modify these expressions.

Vector Properties - Only when both magnitude and direction are exactly the same! - These must be added vectorally! Draw first vector. Draw second vector from tip of first vector (repeat for each additional vector present) Draw resultant (R) from tail of first vector to tip of last vector. - Cumulative law of addition. - Associative law of addition. - Negative of a vector. Same magnitude but opposite direction. - Subtraction of a vector. Same rules as for addition. Multiply a vector by a scalar. Positive scalar changes magnitude. Negative scalar changes magnitude and direction.

Vector Components Vectors can be broken up into components. Components are used to describe part of the vector along each of the coordinate directions for your chosen coordinate system. - The number of components is determined by the number of dimensions. A 2D vector in the x-y plane has 2 components - 1 in the x direction and 1 in the y direction. A 3D vector in spherical coordinates has 3 components - 1 in the r direction, 1 in the q direction and 1 in the f direction. - All vector components are defined to be orthogonal (perpendicular) to each other. - Vector components are also vector quantities. Vectors can be moved as long as their length and orientation do not change! y When you square a vector, the result is a square of the magnitude. Directional information is removed. These are only valid for the specified angle. q x

Vector notation using unit vectors A unit vector is a dimensionless vector with a magnitude of 1 that is used to describe direction. Cartesian coordinate unit vectors: The following notation is also used: x – direction = y – direction = z – direction = x – direction = y – direction = z – direction = - The hat ‘^’ is used to distinguish unit vectors from other vectors. Examples of vectors using unit vector notation: Ax = magnitude of A along x – direction Ay = magnitude of A along y – direction Az = magnitude of A along z – direction x = length along x – direction y = length along y – direction z = length along z – direction

Example: Adding vectors using vector notation You add together magnitudes along similar directions. Remember to include appropriate sign. Rx Ry Rz You can determine magnitude and direction for 3D vectors in a similar fashion to what was done for 2D. qx, qy and qz are the angles measured between the vector and the specified coordinate direction. The magnitude can be found using the Pythagorean Theorem.