Lecture 1eee3401 Chapter 2. Vector Analysis 2-2, 2-3, Vector Algebra (pp ) Scalar: has only magnitude (time, mass, distance) A,B Vector: has both magnitude and direction (velocity, force) A, B or A, B Field: function that specifies a particular quantity everywhere in a region. Scalar field: electric potential distribution Vector field: velocity of raindrops Vector magnitude: |A|, |A| Unit vector: a A has a unit magnitude and direction along A

Lecture 1eee3402 In Cartesian coordinates: A x ; A y ; A z are called the components of A in the x, y, z directions. a x ; a y ; a z are unit vectors in the x, y, z directions. The magnitude of vector A is given by and the unit vector along A is given by

Lecture 1eee3403 Vector additions Basic laws commutative associative distributive

Lecture 1eee3404 Example: 1) 2) 3) A unit vector along Position and Distance vectors P = ( x, y, z ) is a point in Cartesian coordinates Position vector:

Lecture 1eee3405 Distance vector: Example: P = (0, 2, 4); Q = (-3, 1, 5) 1) Position vector P 2) Distance vector from P to Q 3) Distance between P and Q 4) A vector parallel to PQ with magnitude of 10 Example: Sketch the vector field

Lecture 1eee Vector Multiplication There are two types of vector multiplication: 1) Scalar (dot) product: 2) Vector (cross) product: plus 3) Scalar triple product: 4) Vector triple product: Dot Product If

Lecture 1eee3407 then The angle between A and B can be calculated as The dot product obeys Commutative law: Distributive law:

Lecture 1eee3408 Cross Product 1) right-handed screw rule 2) right-hand rule if then

Lecture 1eee3409 Determining the component of a vector: The scalar component A B of A along vector B The vector component A B A - A B is perpendicular to B Example: find the angle between A and B

Lecture 1eee34010 Cross Product Properties: 1) anticommutative 2) it is not associative 3) distributive Scalar Triple Product

Lecture 1eee34011 if then the result is a scalar, it is the volume of a parallelepiped. Vector Triple Product

Lecture 1eee34012 Example: find: 1) 2) 3) 4) 5) 6) A unit vector perpendicular to both Q and R 7) The component of P along Q