Vectors An Introduction

There are two kinds of quantities… Vectors are quantities that have both magnitude and direction (e.g., displacement, velocity, acceleration). Scalars are quantities that have magnitude only (e.g., position, speed, time, mass).

Notating vectors Vector: R R R This is how you draw a vector. head  This is how you draw a vector. R head tail

Notating scalars Scalar: R There is no standard way to draw a scalar!

Direction of Vectors A  x  x B

Vector angle ranges     II 90o <  < 180o I x y II 90o <  < 180o   I 0 <  < 90o III 180o <  < 270o  IV 270o <  < 360o 

Magnitude of Vectors The best way to describe the magnitude of a vector is to measure the length of the vector. The length of the vector is proportional to the magnitude of the quantity it represents.

Magnitude of Vectors A B If vector A represents a displacement of three miles to the north… B Then vector B, which is twice as long, would represent a displacement of six miles to the north!

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

Inverse Vectors Inverse vectors have the same length, but opposite direction. A -A

Vectors: x-component Ax = A cos  Ax A  x

Vectors: y-component Ay = A sin  A  x Ay

Vectors: angle y  = tan-1 (Ry/Rx) Ry  x Rx

Vectors: magnitude y R R = √ (Rx2 + Ry2) Ry x Rx

Graphical Addition of Vectors B A 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 cancelled out by adding it to its inverse, which is called the equilibrant.

Graphical Addition of Vectors B A E R A + B = R E is called the equilibrant vector!

Component Addition of Vectors Resolve each vector into its x- and y-components. Ax = Acos Ay = Asin Bx = Bcos By = Bsin Cx = Ccos Cy = Csin etc. Add the x-components (Ax, Bx, etc.) together to get Rx and the y-components (Ay, By, etc.) to get Ry.

Component Addition of Vectors Calculate the magnitude of the resultant with the Pythagorean Theorem (R = Rx2 + Ry2). Determine the angle with the equation  = tan-1 Ry/Rx.

Relative Motion Vs Vw Vt = Vs + Vw Vw

Relative Motion Vs Vt = Vs + Vw Vw Vw

Relative Motion Vw Vs Vt = Vs + Vw Vw