Vectors An Introduction

There are two kinds of quantities… Scalars are quantities that have magnitude only, such as speed time mass Vectors are quantities that have both magnitude and direction, such as displacement velocity acceleration

R R R Notating vectors head tail This is how you notate a vector… This is how you draw a vector… R R R head tail

A  Direction of Vectors x Vector direction is the direction of the arrow, given by an angle. This vector has an angle that is between 0o and 90o. A  x

Magnitude of Vectors The best way to determine the magnitude (or size) of a vector is to measure its length. The length of the vector is proportional to the magnitude (or size) of the quantity it represents.

Sample Problem If vector A represents a displacement of three miles to the north, then what does vector B represent? Vector C? B A C

Equal Vectors Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity). Draw several equal vectors.

Inverse Vectors Inverse vectors have the same length, but opposite direction. Draw a set of inverse vectors. A -A

Graphical Addition of Vectors

Properties of Vectors Can be moved parallel to themselves in a diagram.

Properties of Vectors Vectors can be added in any order.

Properties of Vectors To subtract a vector, add it’s opposite.

Properties of Vectors Multiplying or dividing vectors by scalars = new vector.

Graphical Addition of Vectors Add vectors A and B graphically by drawing them together in a head to tail arrangement. Draw vector A first, and then draw vector B such that its tail is on the head of vector A. Then draw the sum, or resultant vector, by drawing a vector from the tail of A to the head of B. Measure the magnitude and direction of the resultant vector.

Practice Graphical Addition B A B R A + B = R R is called the resultant vector!

The Resultant and the Equilibrant The sum of two or more vectors is called the resultant vector. The resultant vector can replace the vectors from which it is derived. The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant.

The Equilibrant Vector A + B = R The vector -R is called the equilibrant. If you add R and -R you get a null (or zero) vector.

Graphical Subtraction of Vectors Subtract vectors A and B graphically by adding vector A with the inverse of vector B (-B). First draw vector A, then draw -B such that its tail is on the head of vector A. The difference is the vector drawn from the tail of vector A to the head of -B.

Practice Graphical Subtraction A - B = C

Practice Problem Vector A points in the +x direction and has a magnitude of 75 m. Vector B has a magnitude of 30 m and has a direction of 30o relative to the x axis. Vector C has a magnitude of 50 m and points in a direction of -60o relative to the x axis. Find A + B Find A + B + C Find A – B.

a)

b)

c)