.
A standard vector is a vector in standard position, which
means a vector with initial point at the origin in the Cartesian coordinate
system. Every vector in the plane is equivalent to a standard vector.
A vector in the plane is a directed line segment (represented by an
arrow) from some initial point P to some terminal point Q. Two vectors
are considered equivalent if they have the same length and the same
direction. When the initial and terminal points are named with letters
such as P and Q, the vector can be represented as PQ with an arrow over
the top, i. e. .
A standard vector is a vector in standard position, which
means a vector with initial point at the origin in the Cartesian coordinate
system. Every vector in the plane is equivalent to a standard vector.
Displacement is an example of a quantity measured by a vector. Suppose an object in a plane (the floor, for example) is moved three units to the east and four unit north. Then it would have been displaced a total of five units in an approximately north easterly direction along the diagonal of a right triangle with sides three and four. In the science of Physics, force is a quantity measured by a vector. Every force has a magnitude and a direction of application. The 'length' of a force vector equals the magnitude of the force.
Consider how one would 'add' two displacements. Suppose an object is moved 2 units west, then one unit south. Next, it is moved 3 units east and 4 units north. The sum or result of the two displacements would be the same as if the object had been moved 1 unit east and 3 units north. Picture the first displacement as an arrow going from the initial position of the object to its next position. Then think of the second displacement as an arrow going from the second position of the object to the third position. Then the resulting displacement would be an arrow going from the first position of the object to the third position of the object. See the diagram:
We see that the addition of vectors can be represented by placing the
initial point of the second vector at the terminal point of the first vector,
then the sum of the two vectors is the vector beginning at the initial
point of the first vector and ending at the terminal point of the second
vector. This is sometimes called the triangle rule for the addition of
vectors. Using this same example, we can introduce the idea of the components
of a vector. Vector components are sometimes indicated by an ordered pair
of numbers between two 'angle brackets'. Thus, the first displacement in
this example--two units vest and one unit south--could be represented by
the components < -2, -1 >. The second displacement--three units east
and four units north--could be represented by the components < 3, 4
>. Notice that it is easy to find the components of the sum or resultant
vector--just add the components of the vectors. < -2, -1 > + < 3,
4 > = < 1, 3 >.
Now, consider a standard vector v = < a, b >. (We'll denote vectors with lower case bold letters.) The length or magnitude of of v is denoted by | v | and computed using the Pythagorean Theorem.
Notice that a / | v | = sin θ and b / | v | = cos θ , where θ is the angle between v and the positive x axis.
So v = < a, b > = < | v | sin θ , | v | cos t > = | v | < sin θ , cos θ >. That is,
v = | v | < sin θ , cos θ >.
The vector < sin θ , cos θ > is called the direction of v.
Notice that we factored out the | v | from the components of the vector. We can do this because multiplying a vector by a number changes the length of the vector accordingly. If we multiply a vector by 2, the length is doubled and the direction is unchanged. If we multiply a vector by one-half, the length is halved and the direction is unchanged. However, if we multiply by - 1, the length is unchanged and the direction is reversed.
In general, if v = < a, b >, the c v = < c a, c b >
Exercise 6.4.1
Find the magnitude of the direction vector < sin θ , cos θ >.
Direction vectors are sometimes called unit vectors because they have unit length.
Exercise 6.4.2
Given that v = < -2, 2 >, find the magnitude, the direction angle θ and the direction vector. Find the components of 3 v, and -2 v.
Two special unit vectors are sometimes defined:
i = < 1, 0 > corresponding to the direction angle 0o and
j = < 0, 1 > corresponding to the direction angle 90o.
Every vector in the plane can be represented as a linear combination of the unit vectors i and j.
< a, b > = < a, 0 > + < 0, b > = a < 1, 0 > + b < 0, 1 > = a i + b j
Thus, a i + b j is an alternate way of writing a vector in component form.
Exercise 6.4.3
Find the components of a vector of magnitude 4 and direction angle 45o.
Sometimes vectors are represented solely in terms of their magnitudes and directions, with specifying their components.
Consider the following problem:
Two vectors u and v of magnitude of 5 and 2 respectively are separated by an angle of 30o. Find the magnitude of their sum u + v and the angle the sum makes with the first vector.
The following is a geometrical diagram of the problem:
The vectors u and v are initially drawn from the same initial point so that we can show the 30o separation. A copy of u is then moved so that it begins at the terminal point of v. This new copy of u makes an angle of 180o - 30o = 120o with v. This gives us a triangle with sides 2 and 5 and an included angle of 120o. Thus, we can find the magnitude of u + v using the Law of Cosines.
| u + v | 2 = 52 + 22 - 2 ( 5 ) ( 2 ) cos 150o = 29 + 20 cos 30o = 46.3205 approximately.
So | u + v | is approximately 6.806.
To find the angle θ the sum makes with u, note that it is congruent to the angle opposite v in the triangle. Thus using the Law of Cosines again,
cos θ = 0.9891. Thus, the angle is approximately 8.45o.
Exercise 6.4.4
Two vectors u and v of magnitude of 3 and 2 respectively are separated by an angle of 45o. Find the magnitude of their sum u + v and the angle the sum makes with the first vector.