Scalars and Vectors.

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Presentation transcript:

Scalars and Vectors

A scalar is a number which expresses quantity A scalar is a number which expresses quantity. Scalars may or may not have units associated with them. Examples: mass, volume, energy, money A vector is a quantity which has both magnitude and direction. The magnitude of a vector is a scalar. Examples: Displacement, velocity, acceleration, electric field

Vector Notation Vectors are denoted as a symbol with an arrow over the top Vectors can be written as a magnitude and direction

Vector Representation Vectors are represented by an arrow pointing in the direction of the vector The length of the vector represents the magnitude of the vector WARNING!!! The length of the arrow does not necessarily represent a length

Vector Addition Adding Vectors Graphically The resultant is drawn from the tail of the first to the head of the last vector Arrange the vectors in a head to tail fashion

Vector Addition This works for any number of vectors

Vector Addition

Vector Subtraction Subtracting Vectors Graphically Flip one vector. Then proceed to add the vectors The resultant is drawn from the tail of the first to the head of the last

Vector Components Any vector can be broken into components along the x and y axes Example: from the horizontal. Find its components.

Vector Addition by Components You can add vectors by adding the components of the vector along each direction. Note that you can only add components which lie in the same direction. Never add the x-component and the y-component

Unit Vectors Unit vectors have a magnitude of 1. They only give direction. A displacement of 5m in the x-direction is written as The magnitude is 5m The direction is the direction

Finding the Magnitude and Direction

Vector Multiplication I : The Dot Product The result of a dot product of two vectors is a scalar!

Vector Multiplication : The Dot Product

Vector Multiplication II : The Cross Product The result of a cross product of two vectors is a new vector!

Vector Multiplication II : The Cross Product

Vector Multiplication II : The Cross Product

Vector Multiplication II : The Cross Product