Vectors AP Physics C

Vectors and Scalars A quantity that is fully described by a single number is called a scalar quantity (i.e., mass, temperature, volume). A quantity having both a magnitude and a direction is called a vector quantity. The geometric representation of a vector is an arrow with the tail of the arrow placed at the point where the measurement is made. We label vectors by drawing a small arrow over the letter that represents the vector.

Properties of Vectors Vectors have two properties; magnitude and direction. Magnitude is the size of the vector (the length of the vector). Direction is the direction that the arrow points. This can be expressed by using positive and negative signs or by using angles. If two vectors have the same magnitude and direction, they are the same vector, regardless of where the are located.

Vector Addition Vectors can be added together using what is called the “tip-to-tail” method. The box below shows the steps involved with the method. We will do most vector addition with vectors that are co-linear (lying in the same straight line) or vectors that are perpendicular.

Adding Collinear Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? + 54.5 m, E 30 m, E Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION. 84.5 m, E

Adding Collinear Vectors VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E  30 m, W 24.5 m, E

Adding Perpendicular Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. 55 km, N 95 km,E

Adding Perpendicular Vectors BUT…..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. To find the value of the angle we use a Trig function called TANGENT. 109.8 km 55 km, N q N of E 95 km,E

Scalar Multiplication Multiplying a vector by a positive scalar gives another vector of different magnitude but pointing in the same direction. A vector cannot have a negative magnitude. If we multiply a vector by a negative number we reverse its direction. Multiplying a vector by –1 reverses its direction without changing its length (magnitude).

Vector Multiplication Direction and the method of multiplication matter when multiplying vectors. In the end, the resultant is dependent on the angle between the initial vectors. Method #1: The dot product The dot product of two vectors returns a scalar (because math [it’s really not important]). Method #2: The cross product The cross product of two vectors returns a new vector (also because math [for real]).

Coordinate Systems A coordinate system is an artificially imposed grid that you place on a problem. You are free to choose: Where to place the origin, and How to orient the axes (you can rotate the axis if you wish, which is sometimes beneficial). To the right is a conventional xy-coordinate system and the four quadrants I through IV. We will discuss the most appropriate choices of coordinate system for each problem that we encounter in this class.

Vector Components In many units we will need to break a vector into components. This is the reverse process of vector addition (this is often called vector decomposition). You can break a vector into as many pieces as you want, be we generally just want one component that is parallel to our x-axis (we call this our “x” component) and one component that is parallel to the y axis (we call this our “y” component).

Tilted Axis For motion on a slope, it is often most convenient to put the x-axis along the slope. When we add the y-axis, this gives us a tilted coordinate system. Finding components with tilted axes is done the same way as with horizontal and vertical axes. The components are parallel to the tilted axes and the angles are measured from the tilted axes.

Vector Decomposition To accomplish vector decomposition, we will use basic right triangle trig. This will be a very important tool in all of physics, particularly with projectiles and forces.

Example Determine the x and y components of the velocity vector below. In addition, determine the magnitude and direction of the velocity vector.

Example Find the x- and y-components of the acceleration vector a shown below.

Common Conventions We will often represent the direction of vectors and vector components by using positive and negative signs. Conventionally, a vector or vector component that points either up or to the right is considered positive, and if it points down or to the left it is negative. This matches the traditional xy-coordinate system. You do not have to adopt this convention, however you must be consistent. If you say that a vector is positive, then a vector pointing in the opposite direction must be negative.

Unit Vectors Each vector in the figure to the right has a magnitude of 1, no units, and is parallel to a coordinate axis. A vector with these properties is called a unit vector. These unit vectors have the special symbols: Unit vectors establish the directions of the positive axes of the coordinate system. When decomposing a vector, unit vectors provide a useful way to write component vectors: