L 2 – Vectors and Scalars Outline Physical quantities - vectors and scalars Addition and subtraction of vector Resultant vector Change in a vector quantity, calculating relative and resolve a vector into components Vector representation in a component form in a coordinate system

Vectors in Physics-Examples Many physical quantities have both magnitude and direction: they are called vectors. Examples: displacement, velocity, acceleration force, momentum... Other physical quantities have only magnitude: they are called scalars. Examples: distance, speed, mass, energy...

Displacement and Distance Displacement is the vector connecting a starting point A and some final point B A B Distance is the length one would travel from point A to the final point B. Therefore distance is a scalar

Geometrical Representation of Vectors Arrows on a plane or space To indicate a vector we use bold letters or an arrow on top of a letter

Properties of Vectors 1.The opposite of a vector a is vector - a 2.It has the same length but opposite direction 3.Two vectors a and b are parallel if one is a positive multiple of the other: a = m b, m>0 Example: if a = 3 b, then a is parallel to b (if a = -2 b then a is anti-parallel to b)

Operations: Adding two Vectors When we add two vectors, we get the resultant vector a + b, with the parallelogram rule:

Operations: Adding more vectors We can add more vectors by pairing them appropriately

Operations: Vector Subtraction Special case of vector addition  Add the negative of the subtracted vector a – b = a + (– b)

Components of a Vector A component is a part or shadow along a given direction It is useful to use rectangular components –These are the projections of the vector along the x- and y-axes

Components of a Vector, cont. The x-component of a vector is the projection along the x-axis  a x = a cos θ The y-component of a vector is the projection along the y-axis  a y = a sin θ a is the magnitude of vector a  a 2 = a x 2 + a y 2

Example 1 Resolve this vector along the x and y axes to find its components respectively.

Example 2 A vector of 15.0 N at 120º to the x-axis is added to the vector in Example 1. Find the x and y components of the resultant vector N 10.0 N

The Unit Vectors: i, j, k A unit vector has a magnitude of 1 i is the unit vector in the x-direction, j is in the y-direction and k is in the z-direction.

The Unit Vectors, Magnitude Example 3 : Given the two displacements Show that the magnitude of e is approximately 17 units where: Any vector a can be written as:  a = x i + y j + z k

S - South N - North E - East W - West 15° 15 ° east of north, or 75 ° north of east, or bearing of 15 ° 45 ° west of south, or 45 ° south of west, or bearing of 225 ° 45° 30° …..? ….? 30° Direction of Vectors

Example 4 Find the magnitude and direction of the electric field vector E with components 3i – 4j. Note: This vector could also be written in matrix form:

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by :  V CP = V CE - V PE Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity V CE. A pedestrian (P) is travelling relative to the earth (E) with velocity V PE. Relative velocity

Example 5 A boat is heading due north as it crosses a wide river with a velocity of 8.0 km/h relative to water. The river has a uniform velocity of 6.0 km/h due east. Determine the velocity (i.e. speed and direction) of the boat relative to an observer on the riverbank.

The dot (scalar) product Imagine two vectors a, b at an angle θ The dot product is defined to be:  a · b = a b cos θ  Useful in finding work of a force F

CHECK LIST READING Serway’s Essentials of College Physics pages and Adams and Allday: 3.3 pages 50-51, Summary Be able to give examples of physical quantities represented by vectors and scalars Understand how to add and subtract vectors Know what a resultant vector is Know how to find the change in a vector quantity, calculate relative and resolve a vector into components Understand how vectors can be represented in component form in a coordinate system Be able to do calculations which demonstrate that you have understood the above concepts

Numerical Answers for Examples Ex 1 – Vx = 8.7N, Vy = 5N Ex 2 – coordinates of resultant vector are (1.16, 18.0) Ex 3 – length is 16.9 units, or approximately 17 units Unknown Directions – Yellow is 60° S of E, or 30 ° E of S, or bearing 150° (90+60) Blue is 30° N of W, or 60 ° W of N, or bearing 300° (270+30) Ex 4 – Resultant vector E : magnitude 5 units, –Direction 53° S of E, or 37° E of S, or bearing 143° ( ) Ex 5 – velocity of boat relative to earth: magnitude 10 km/hr –Direction 53° N of E, or 37° E of N, or bearing 37°