Introduction Vectors Vectors Vector algebra forms the mathematical foundation for kinematics and dynamics. Geometry of motion is at the heart of both the kinematics and the dynamics of mechanical systems. Vector analysis is the time-honored tool for describing geometry. This presentation provides a brief overview of vectors and vector algebra. Full understanding of vector algebra is necessary in order to perform kinematic and dynamic analysis of mechanical systems.

Vector definition Vectors Vector definition A vector is presented as a directed line with a known length (magnitude). A vector is geometrically presented as an arrow. Colors and different line widths are only used for visual effects. The starting point of a vector is also called the tail, and the end point is also called the tip. starting point (tail) end point (tip)

Vector naming Vectors Naming vectors Vectors often have names. To distinguish between vectors and scalars, we use bold upper case characters for vectors and light face characters for scalars. Subscripts normally represent the points that a vector connects. For example, PBA is a vector connecting point B to point A, where the tail is at A and the tip is at B. We often use numbers to distinguish between vectors of the same kind. For example, velocity vectors V1, V2, V3 … A A PBA B V1 V2 V3

Vector length Vectors Vector length The length of a vector, which is also called magnitude, is denoted by the same character as the name of the vector, but not in bold. For example the magnitude of R is R or R (it is also shown as |R|). The table below shows the lengths of the vectors that appear on the right (arbitrary units). A B C A = 1.46 B = 2.2 C = 0.45 D = 1.46 D

Vector direction Vectors 232° Vector direction The direction of a vector is usually described by an angle. We always measure the angle between the vector and an horizontal line starting at the tail of the vector pointing to the right (positive x-axis). Counter-clockwise (ccw) direction is defined as positive. This definition will simplify the process of projecting a vector unto coordinate axes as will be seen later! Examples on the right illustrate this definition. 50° 180° −73° 336° 156°

Vector construction We summarize our notation for length and angle as: Vectors Vector construction We summarize our notation for length and angle as: length @ angle For example R = 3.7 @ 302°. This gives us all the information needed to draw R: 1. Select the starting point 2. Draw an horizontal line Draw a second line at an angle of 302° Measure the length for 3.7 units 5. Complete the vector 302° ► ► 3.7 ► ► ►

Vector addition Vectors Vector addition To add two vectors A and B, we draw vector B so that its starting point coincides with the tip of vector A. We then draw the resultant vector from the starting point of A to the tip of B. Note that: A + B = B + A Example: A = 2.4 @ 200° B = 4 @ 72° 1. Draw A 2. Select the tip of A as the:starting point of B. Draw B 3. Draw the resultant vector from the starting point of A:to the tip of B A+B 4 B 200° ► A 72° 2.4 ► ►

Scalar multiplication Multiplication with a scalar Vectors Scalar multiplication A vector can be multiplied by a scalar (number). If a vector is multiplied by a positive number, the length is multiplied by that number. The angle remains the same. If a vector is multiplied by a negative number, the length is multiplied by the absolute value of the number. The angle is increased by 180° (reversing the direction of the vector). Examples: A 1.7 A 0.4 A -0.4 A -1.3 A

Vector subtraction Vectors Vector subtraction We can use the scalar product and vector addition to subtract two vectors. For example A − B is the same as A + (−1) B. Example: A = 4 @ 233° B = 7.4 @ 300° 1. Draw A 2. Select the tip of A as the starting point of −B. Draw:−B (300° + 180° = 480° =:120°) 3. Draw the resultant vector;from the starting point of A;to the tip of −B A−B 233° −B 7.4 ► 4 A ► 120° ►