- Graphical Representation as Arrows
Sometimes vectors are represented graphically by an arrow. The magnitude of the vector is the length of the arrow, and the direction is the direction that the arrow is pointing.Given points
and 
we could ask for the vector from 
to 
, 
. You can draw the vector aw an arrow beginning at 
and ending at 
.
You should remember, however, that the starting point is not part of the definition of a vector, so any other vector with the same magnitude and direction is equal to
, even if you don't draw it starting at 
.
- Components
We can also represent a vector as a pair of numbers representing its horizontal and vertical components,
. This notation tells you that the vector moves horizontally by 
and vertically by 
.
This notation is similar to the notation for coordinates of a point,
. In fact, the vector from the origin to a point 
is 
. More generally, the vector from point 
to point 
is 
.
The magnitude of a vector
is denoted 
and can be found by drawing a right triangle and using the Pythagorean Theorem. If the triangle has horizontal side 
and vertical side 
then the length of the hypoteneuse is
It's a little trickier to determine the direction angle of a vector in component form.
does not change the direction of the vector, but multiplies its magnitude by 
.
multiplies the magnitude by 
and changes the direction by 
. This means that 
is the vector with the same magnitude as 
, but pointing in the opposite direction.
ought to be a vector such that if you add it to 
you get 
, since 
.
So if you draw 
and 
starting from the same point, then 
will be the vector that goes from the head of 
to the head of 
.
.
If 
then 
, so
.
and 
.
and 
like this:
you can get a unit vector pointing in the same direction a 
by rescaling 
so that is has magnitude 1. Check that 
has magnitude 1.
you can find the unit vector pointing in that direction by taking the vector from the orign to the point 
on the unit circle. We know that the coordinates of the pont 
on the unit circle are 
, so the vector we get is 
.
As a combination of the standard unit vectors this is 
.
.
The unit vector pointing in direction 
is \left<cos 30ˆ, sin 30ˆ\right>;. Multiply this vector by 2 to give it magnitude 2.
So 
is the vector with magnitude 2 pointing in directin 
. In compnents this would be
.
and magnitude 4 in components.
is the unit vector poining in direciton 
. Mulitply by 4 to get the right magnitude.
and magnitude 4 as a combination of standard unit vectors.
.
is
and 
with vertical. Find the tension in each of the ropes. 
and 
, so 
.
. How much force do you need to pull with, ignoring friction?
.
Split 
into components parallel and perpendicular to the ramp, 
and 
.
, angle 
, and angle 
.
, so
and 
are unit vectors.
.
.
We also know that the dot product of a vector with itself is the magnitude squared, so
and 
is the same as the angle between 
and 
, so the angle between 
and 
satisfies
and 
. 
incline.
We needed to split the force of gravity pulling the boat down into two components, one pulling the boat perpendicularly into the ramp and another pulling the boat down the ramp.
In that example we knew the angle of the ramp, so we could find a triangle and all its angles and the length of one of the sides.
and a vector 
.
that points in the 
direction.
This will be the vector that makes bottom side of the right triangle below:
.
We also know its direction, it is the same as the direction of 
.
So:
is a unit vector. If it is not then we should use the unit vector that points in the same direction:
. How much force do you need to pull with, ignoring friction?
.
inline, so a unit vector pointing up the ramp is 
. The componnent of 
parallel to the ramp is:
.