Vectors
Vectors are usually used to represent velocity and acceleration, force, and other
directional quantities in physics.
Vectors are quantities with size and direction.
The objects that we have worked with in single variable calculus (Calculus 1 and
2) have all had a quantity, i.e. we were able to measure them.
Some quantities only have size, such as time, temperature, or weight. These quantities
are called scalars. Other quantities can have size and direction.
Velocities, for example, have direction as well, and therefore they are described
as vectors. We denote vectors with an arrow pointing in the direction they are oriented.
The direction of a vector on the coordinate plane is intuitive. The positive y direction,
which is up, is north, and the positive x direction is east. The following vector
is slightly east of north.
The direction of a vector can also be described with a quantity. Usually, the direction
of vectors are stated in relation to another direction. The following vector is
described as “5 miles per hour 53.13 degrees north of east.”
This vector can also be described as “5 miles per hour 36.87 degrees east of north.”
To simplify the values of vectors, we use the x axis (or east) as a starting point
for measurement. A line lying on the x axis would have a direction of 0 degrees.
The following vector can be denoted with many different directions.
The last vector would be 53.13 degrees south of west.
Scalars and Vectors
Remember that scalars only have size, while vectors have size and direction.
Speed and velocity are different too. While they are sometimes used interchangeably,
speed is considered a scalar while velocity is considered a vector.
There is a distance between distance and displacement as well. Distance is a scalar
because it only has size. Displacement, however, is a vector because it tells us
how far an object moved in a certain direction.
Scalars can be manipulated by the laws of arithmetic for real numbers, while vectors
have special laws that need to be followed when operating on each other. For instance,
if you walked 4 blocks and then 3 more blocks, how many blocks have you walked?
We can add these quantities together to get 7 blocks. However, if you walked 4 blocks
east and 3 blocks north, how far from your starting point will you have walked?
Since these vectors are in different directions, we cannot simply add them together.
The amount of degrees traveled can be either measured from the image or calculated
using trigonometry.
The resulting vector would be 5 blocks at .644 radians.
Vector Notation
Vectors have a special notation that distinguish them from scalars. Vectors can
be noted as
For our purposes, we will always denote a vector with an arrow on the top to denote
a quantity with direction.
The previous vector would be denoted as
We can also use unit vectors i and j to denote a vector where i = <1,0>
and j = <0,1>
Magnitude, or length of the vector is denoted as
We use the magnitude to find the quantity of the vector. Whenever we want to disregard
the direction of a vector (taking the area, volume, etc), we can juse take the magnitude.
The direction of the vector is denoted as
Vector Equalities and Operations
Equal Vectors
Have the same magnitude and the same direction, they do not need to have the same
starting points.
Opposite Vectors
Have exactly the same length but point in the opposite direction. When added together,
opposite vectors cancel each other out.
Parallel Vectors
Have the same direction but different lengths.
Vectors that have the same direction can be multiplied by scalars to yield a different
magnitude.
Vector Addition
When adding vectors, we attach the start of the second vector (intial point) to
the end of the first vector (terminal point).
Vector Subtraction
Scalar Multiplication
Scalar multiplication is when a vector is multiplied by a scalar to increase or
decrease the magnitude of the vector. The scalar does not have any effect on the
direction of the vector.
Dot Product
If we have two vectors u and v, the dot product is denoted as
where |u| and |v| are the magnitudes and Θ is the angle between the vectors.
To illustrate what the dot product means, let’s take the last part of the formula
and deconstruct it.
If we take the vector v times the cos(Θ), we will end up with
the projection of v onto u. The projection is formed by dropping a
perpendicular line from the terminal point of v onto u, therefore forming a right
angle. The projection of v onto u is the amount of vector v going in the u direction.
The dot product of v and u just multiplies the projection of v and the vector u
(or vice versa).
If we go back to our formula, we can substitute the projection of v for the vector
v.
This result tells us how much of vector v is going in the direction of vector
u.
What is this useful for? If we think about physics applications, if we have two
forces at an angle, we can see how much force is going in a particular direction.
The dot product is sometimes called the scalar product because it always yields
a scalar quantity. The dot product can also help us measure the angle between vectors,
find projections, and determine if two vectors are perpendicular, as we will see
in the next examples.
Note that perpendicular vectors will always yield a dot product of 0 because there
is no projection, i.e. no amount of either vector going in the other vector’s direction.
The dot product can also be notated with unit vectors i = <1,0> and j = <0,1>
where a, b, c, and d are constants.
Angle Between two Vectors
We can use the definition of the dot product to find the angle between any two vectors.
All we need to do is isolate Θ
The Projection of a Vector
From reworking the Dot Product formula and dividing by |u|
we can conclude that the projection of v onto u is