Summary
Most things we encounter in Physics are either scalars or vectors.
A scalar quantity has only magnitude and no direction.
This means we can fully explain a scalar with a number and a unit.
A vector quantity has both magnitude and direction.
The distinction between scalars and vectors is important. We will use bold face type to indicate a vector, such as r. In writing a vector by hand, we will indicate a quantity is a vector by drawing an arrow above it as . Some such distinguising notation is important. Do not write a vector without some distinguishing characteristic or notation.
Consider adding two vectors together as in A + B = C. Each vector can first be resolved into its respective components,
A = (Ax , Ay , Az)B = (Bx , By , Bz)
The components are then added as the scalars they are and the new vector sum is reconstructed,
C = (Cx , Cy , Cz)where
Cx = Ax + BxCy = Ay + By
and
Cz = Az + BzWe will often write this using unit vector notation as
A = Ax i + Ay j + Az kB = Bx i + By j + Bz k
C = Cx i + Cy j + Cz k
where
Cx = Ax + BxCy = Ay + By
and
Cz = Az + Bzor
C = A + BC = (Ax + Bx) i + (Ay + By) j + (Az + Bz) k
Vectors can also be added graphically. A graphical vector addition diagram is an important check on any numerical vector addition.
Return to ToC, Vectors (c) 2005, Doug Davis; all rights reserved