VECTORS A bee will fly from rose to rose To gather up some nectar. And though his path is quite complex His displacement is a vector!
WHAT IS A VECTOR? A vector is a device used when the direction of the parameter being measured is important. Some examples of vectors are: velocity, force, momentum, electric field strength, and torque A vector quantity is typically written in boldface or by placing an arrow over the label. A vector quantity may be represented by drawing an arrow. The length of the arrow is proportional to the magnitude of the vector quantity and the arrow points in the direction of the vector
WHAT IS A VECTOR? The arrow is usually drawn at the origin of a coordinate system and the angle is measured (in radians or degrees) from an axis. Once a vector is drawn, it may be moved to any other location provided the length and direction of the arrow does not change
VECTOR ADDITION (determining a resultant) The maximum resultant occurs when the angle between the vectors being added is 0 and the minimum when the angle is 180 0 Vectors may be added several different ways: 1. Graphically – draw all vector quantities to scale, arrange head to tail, connect the origin of the first vector to the head of the last. (REMEMBER THE BEE) 2. Mathematically Law of Cosines - C 2 = A2 A2 + B2 B2 – 2ABcos (The Theory of Pythagoras is a special case) Components (Two and three dimensions coming soon!)
DETERMINING COMPONENTS Determine the components of a velocity vector with magnitude 20 m/s directed at 37 o above the + x-axis. v =20m/s 37 o vyvy vxvx
EXAMPLE OF VECTOR ADDITION A B C AyAy AxAx CxCx CyCy BxBx ByBy
VECTOR SUBTRACTION To determine the difference between two vectors arrange them with the tails together The direction of the resultant is determined by the order of the equation A B A - BB - A A B
UNIT VECTORS This is a fancy name for a vector of length one (1). A unit vector is constructed by dividing a vector by its magnitude Unit vectors in the x, y, z direction are denoted as: A unit vector with an x component of 6,6, a y component of – 3, and a z component of zero is written as:
UNIT VECTORS - example If: Then the magnitude of A is 6.71 and the unit vector is:
ADDITION IN UNIT VECTOR NOTATION Adding vectors this way is easy!
OTHER FUNCTIONS WITH VECTORS Dot product (scalar product) Used to determine the amount of one vector in the direction of another or the angle between vectors Work is an example of a dot product (W= F. d (“f dot d”) or, Fdcos ) Using unit vector notation:
Dot product example Determine the angle between the following two vectors:
Cross product (vector product) Used to produce a third vector. The result is perpendicular to the first two vectors (See fig. 3 – 20) Torque is an example of a cross product ( =r x F (“r cross F”) or rFsin ) OTHER FUNCTIONS WITH VECTORS
Calculating a cross product using a determinant This method will require practice!! See TACTIC 5 ON PAGE 51.
DOT AND CROSS PRODUCTS http://www.tutorvista.com/content/phy sics/physics-iii/kinematics/scalar- product-animation.php
VECTORS! KEEP THEM STRAIGHT AND YOUR LIFE WILL HAVE DIRECTION!