A vector is commonly drawn as an arrow. The drawing is not decorative. The tail shows the starting point, the tip shows the direction, and the arrow length represents the vector magnitude.
If a force arrow is longer, the force is larger on the same drawing scale. If the arrow points up and to the right, the force acts up and to the right.
Vector notation should keep two pieces of information visible: magnitude and direction.
Two special vector names appear often once vectors are calculated using components.
A zero vector has magnitude 0. Its tail and tip coincide, so its direction is undefined. In physics, a zero vector can appear when two equal and opposite forces cancel each other.
A unit vector has magnitude 1 and only shows direction. Unit vectors do not tell us how strong a force is or how far a displacement is. They identify axis directions.
i^=unit vector in the x direction
j^=unit vector in the y direction
k^=unit vector in the z direction
Later, when a vector is written as A=Axi^+Ayj^, i^ and j^ give the directions, while Ax and Ay give the component magnitudes along each axis.
The first mistake is treating AB as the same vector as BA because they use the same two points. Their directions are opposite.
The second mistake is writing a vector magnitude as a negative number. A magnitude is always 0 or positive. A negative sign on a vector usually means the direction is opposite to a chosen reference direction, not that the length is negative.
For example, −a does not mean the vector has negative magnitude. It means the vector has the same magnitude as a but points in the opposite direction.
