Vector:
A vector is a quantity that has both magnitude and direction. (Magnitude just means 'size'.)
Examples of Vector Quantities:
· I travel 30 km in a Northerly direction (magnitude is 30 km, direction is North - this is a displacement vector)
· The easterly wind is 3m/sec.
Other examples of vectors include:
Acceleration, momentum, queue, angular momentum, magnetic and electric fields
Each of the examples above involves magnitude and direction.
Note: A vector is not the same as a scalar. Scalars have magnitude only. For example, a weight of 50kg is a scalar quantity, since no direction is given. Other examples of scalar quantities are:
Volume, density, temperature, mass, speed, time, length, distance, work and energy.
Each of these quantities has magnitude only, and do not involve direction.
We will use a bold capital letter to name vectors. For example, a force vector could be written as F.
Alternative vector notations
· Some textbooks write vectors using an arrow above the vector name, like this:
· You will also see vectors written using matrix-like notation. For example, the vector acting from (0, 0) in the direction of the point (2, 3) can be written [2 3]
A vector is drawn using an arrow. The length of the arrow indicates the magnitude of the vector. The direction of the vector is represented by (not surprisingly :-) the direction of the arrow.
Example 1 - Vectors
The displacement vector A has direction 'up' and a magnitude of 4 cm.
Vector B has the same direction as A, and has half the magnitude (2 cm).
Vector C has the same magnitude as A (4 units), but it has different direction.
Vector D is equivalent to vector A. It has the same magnitude and the same direction. It doesn't matter that A is in a different position to D - they are still considered to be equivalent vectors because they have the same magnitude and same direction. We can write:
A = D
Note: We cannot write A = C because even though A and C have the same magnitude (4 cm), they have different direction. They are not equivalent.
Free and Localized Vectors
So far we have seen examples of "free" vectors. We draw them without any fixed position.
Another way of representing vectors is to use directed line segments. This means the vector is named using an initial point and a terminal point. Such a vector is called a "localized vector".
Example 2 - Localized Vectors
A vector OP has initial point O and terminal point P. When using directed line segments, we still use an arrow for the drawing, with P at the arrow end. The length of the line OP is an indication of the magnitude of the vector.
We could have another vector RS as follows. It has initial point R and terminal point S.
Because the 2 vectors have the same magnitude and the same direction (they are both horizontal and pointing to the right), then we say they are equal. We would write:
OP = RS
Note that we can move vectors around in space and as long as they have the same vector magnitude and the same direction, then they are considered equal vectors.
Magnitude of a Vector
We indicate the magnitude of a vector using vertical lines on either side of the vector name.
The magnitude of vector PQ is written |PQ|.
We also used vertical lines like this earlier in the Numbers chapter (where it was called 'absolute value', a similar concept to magnitude).
So for example, vector A above has magnitude 4 units. We would write the magnitude of vector A as:
| A | =4
Scalar Quantities
A scalar quantity has magnitude, but not direction.
For example, a pen may have length "10 cm". The length 10 cm is a scalar quantity - it has magnitude, but no direction is involved.
Scalar Multiplication
We can increase or decrease the magnitude of a vector by multiplying the vector by a scalar.
Example 3 - Scalar Multiplication
In the examples we saw earlier, vector B (2 units) is half the size of vector A (which is 4 units) . We can write:
B = 0.5 A
This is an example of a scalar multiple. We have multiplied the vector A by the scalar 0.5.
Example 4 - Scalar Multiplication
Vectors in Opposite Directions
We have 2 teams playing a tug-of-war match. At the beginning of the game, they are very evenly matched and are pulling with equal force in opposite directions. We could name the vectors OA and OB.
Zero Vectors
A zero vector has magnitude of 0. It can have any direction.
A vector may have zero magnitude at an instance in time. For example, a boat bobbing up and down in the water will have a positive velocity vector when moving up, and a negative velocity vector when moving down. At the instant when it is at the top of its motion, the magnitude is zero.
In the tug-of-war example above, the teams are evenly matched at a certain instant and neither side is able to move. In this case, we would have:
OA + OB = 0
The 2 force vectors OA and OB, operating in opposite directions, cancel each other out.
Unit Vectors
A unit vector has length 1 unit and can take any direction.
A one-dimensional unit vector is usually written i.
Example 5 - Unit Vector
In the following diagram, we see the unit vector (in red, labeled i) and two other vectors that have been obtained from i using scalar multiplication (2i and 7i).
Vectors in 2 Dimensions
On this page...
Components of Vectors
Magnitude of a Vector
Direction of a Vector
So far we have considered 1-dimensional vectors only.
Now we extend the concept to vectors in 2-dimensions. We can use the familiar x-y coordinate plane to draw our 2-dimensional vectors.
The vector V shown above is a 2-dimensional vector drawn on the x-y plane.
Components of Vectors
Reading from the diagram above, the x-component of the vector V is 6 units.
The y-component of the vector V is 3 units.
We can write these vector components using subscripts as follows:
Vx = 6 units
Vy = 3 units
Magnitude of a 2-dimensional Vector
The magnitude of a vector is simply the length of the vector. We can use Pythagoras' Theorem to find the length of the vector V above.
Recall (from Section 1, Vector Concepts) that we write the magnitude of V using the vertical lines notation | V |.
We have:
Magnitude of V / =|V|=root(6^2+3^2)
=root(45)
=6.71units
Direction of a 2-dimensional Vector
To describe the direction of the vector, we normally use degrees (or radians) from the horizontal, in an anti-clockwise direction.
We use simple trigonometry to find the angle. In the above example, we know the opposite (3 units) and the adjacent (6 units) values for the angle (θ) we need.
So we have:
tanθ=36=0.5
This gives:
θ / = arctan 0.5= 26.6°
(= 0.464 radians)
So our vector has magnitude 6.71 units and direction 26.6° up from the right horizontal axis.
4. Adding Vectors (in 2 dimensions)
Let's first have a play. The following Flash interactive involves a Cessna that is trying to land on the runway, but it is a bit windy.
You are the pilot. If you have lined up the Cessna properly, you'll be able to land. If not, you need to go around and try again. You can only land towards the north (towards the top).
Choose your wind direction and away you go. You can steer with the right and left arrow keys on your keyboard.
Have a fly around and then attempt your landing. The challenge is to put it down exactly in the center of the runway.
Adding Vectors Using a Parallelogram
In the Flash interactive activity above, you would have noticed a parallelogram of forces that changed with the change in heading of the plane or the wind direction.
The parallelogram is an alternative method to using triangles. If we add the the blue (heading) vector and the black (wind) vector the resultant vector is the red ground direction vector. In the image, the ground direction is due North.
Unit Vectors and Components of a Vector (2-D)
We met the idea of a "unit vector" before in 1. Vector Concepts. We now extend the idea for 2-dimensional vectors.
The diagram shows a unit vector in the x-direction (called vector i) and another in the y-direction (called vector j).
We can write any 2-dimensional vector in terms of the unit vectors i and j.
Example
In an earlier example, we had the following vector:
We could write the components of the vector V as follows.
Vx = 6 i
Vy = 3 j
So we can write the vector V using unit vectors as follows:
V=6i+3j
Dot Product (aka Scalar Product) in 2 Dimensions
Also on this page:
If we have any 2 vectors P and Q, the dot product of P and Q is given by:
P ⋅ Q = |P| |Q| cos θ
where
|P| and |Q| are the magnitudes of P and Q respectively, and
θ is the angle between the 2 vectors.
The dot product of the vectors P and Q is also known as the scalar product since it always returns a scalar value.
The term dot product is used here because of the • notation used and because the term "scalar product" is too similar to the term "scalar multiplication" that we learned about earlier.
Example 1
a. Find the dot product of the force vectors F1 = 4 N and F2 = 6 N acting at 40° to each other as in the diagram.
b. Find the dot product of the vectors P and Q if |P| = 7 units and |Q| = 5 units and they are acting at right angles to each other.
The second example illustrates an important point about how scalar products can be used to find out if vectors are acting at right angles, as follows.
Dot Product and Perpendicular Vectors
If 2 vectors act perpendicular to each other, the dot product (ie scalar product) of the 2 vectors has value zero.
This is a useful result when we want to check if 2 vectors are actually acting at right angles.
Dot Products of Unit Vectors
For the unit vectors i (acting in the x-direction) and j (acting in the y-direction), we have the following dot (ie scalar) products (since they are perpendicular to each other):
i ⋅ j = j ⋅ i = 0
Example 2
What is the value of these 2 dot products:
a. i ⋅ i
b. j ⋅ j
Answer
Alternative Form of the Dot Product
Recall that vectors can be written using scalar products of unit vectors.
If we have 2 vectors P and Q defined as:
P = a i + b j
Q = c i + d j,
where
a, b, c, d are constants;
i is the unit vector in the x-direction; and
j is the unit vector in the y-direction,
then it can be shown that the dot product (scalar product) of P and Q is given by:
P ⋅ Q = ac + bd
Answer
Example 3 - Alternative Form of the Dot Product
Find P • Q if
P = 6 i + 5 j and
Q = 2 i − 8 j
Answer
Now we see another use for the dot product − finding the angle between vectors.
Angle Between Two Vectors
We can use the dot product to find the angle between 2 vectors. For the vectors P and Q, the dot product is given by
P ⋅ Q = |P| |Q| cos θ
Rearranging this formula we obtain the cosine of the angle between P and Q:
cosθ=P⋅Q/|P||Q|
To find the angle, we just find the inverse cosine of the expression on the right.
So the angle θ between 2 vectors P and Q is given by
θ=arccos(P⋅Q/|P||Q|)
Example 4
Find the angle between the vectors P = 3 i − 5 j and Q = 4 i + 6 j
The 3-dimensional Co-ordinate System
We can expand our 2-dimensional (x-y) coordinate system into a 3-dimensional coordinate system, using x-, y-, and z-axes.
The x-y plane is horizontal in our diagram above and shaded green. It can also be described using the equation z = 0, since all points on that plane will have 0 for their z-value.
The x-z plane is vertical and shaded pink above. This plane can be described using the equation y=0
The y-z plane is also vertical and shaded blue. The y-z plane can be described using the equation x=0 .
We normally use the 'right-hand orientation' for the 3 axes, with the positive x-axis pointing in the direction of the first finger of our right hand, the positive y-axis pointing in the direction of our second finger and the positive z-axis pointing up in the direction of our thumb.
7. Vectors in 3-D Space
On this page...
Magnitude of a 3-D Vector
Adding 3-D Vectors
Dot Product of 3-D Vectors
Direction Cosines
Angle Between Vectors
Application
We saw earlier how to represent 2-dimensional vectors on the x-y plane.
Now we extend the idea to represent 3-dimensional vectors using the x-y-z axes. (See The 3-dimensional Co-ordinate System for background on this).
Example
The vector OP has initial point at the origin O (0, 0, 0) and terminal point at P (2, 3, 5). We can draw the vector OP as follows:
======
Array: An array is a systematic arrangement of objects, usually in rows and columns.
N-dimensional vector can be expressed by array, and vice versa.
For example, I have your heigh, which I use a variable to denote:
X= [172 180 171 163 163….];
Matrix:
A matrix is a rectangular array of numbers or other mathematical objects, for which operations such as addition and multiplication are defined.[4] Most commonly, a matrix over a field F is a rectangular array of scalars from F.[5][6] Most of this article focuses on real and complex matrices, i.e., matrices whose elements are real numbers or complex numbers, respectively. More general types of entries are discussed below. For instance, this is a real matrix: