Robert and John start walking from the same point. Assume that they are walking at the same speed. Does it necessarily mean that they are at the same place at any point of time?
Unless you confirm that they are walking in the same direction, the answer is, they are not.
If the persons are walking in opposite direction, they are apart from each other as much as twice the distance from the starting point.
This is the basic concept of a vector. A vector indicates the magnitude and also the direction in which the magnitude is applied.
Vector and Scalar
A vector gives the magnitude as well as a direction. A scalar represents only the magnitude of a quantity.
In certain cases, the direction may not have meaning. For example, age of a person, height of a tree, mass of a ball etc. It is sufficient to describe such items with magnitude alone and there is no question of direction in all such cases. Such quantities are called scalars.
In many cases, the magnitude alone will not give a true picture. If a force is to be applied on a body we should know in what direction is to applied to achieve the required objective. If you have to shoot an object, unless you shoot in the required direction, you will not be able to hit the target. The quantities which require a direction in addition to the magnitude for a complete description are called vectors.
Vector Examples
Below you could see vector examples
Solved Examples
Question 1: Which one of the following is a vector?
Options : - 2 hours
- 40 ft-lb
- 40 mi/hr
- 5 lb/ft3
Solution:
2 hours is the measure of a time. Time has no direction. The complete meaning is understood by the magnitude itself. Hence it is a scalar.
40 ft-lb is the measure of a force. Force is always to be specified with the direction in addition to the magnitude. Hence it is a vector.
40 mi/hr is the measure of a speed. Speed can be defined by magnitude alone. Hence it is a scalar.
5 lb/ft3 is the measure of a density. Density is a ratio of the mass of a substance to the volume. Since mass and volume are scalrs, density is also a scalar.
Therefore, the correct choice is 2.
Answer : 40 ft-lb
Question 2: Which one of the following is an incomplete statement?
Options : - Robert is 6 ft tall
- The boiling point of water is 2120 F
- Ship A is 20 miles at an angle of 450 from ship B
- Bob is 17 years old
Solution:
Robert is 5 ft tall, describes a height, a measure of length. There is no direction involved here. Hence the statement is complete.
The boiling point of water is 2120 F. It describes a temperature. It can be defined by magnitude alone. Hence the statement is complete.
Ship A is 20 miles at an angle of 450 from ship B. Unless the direction of the angle specified, you can not figure out exactly the location of ship B. Hence this statement is incomplete.
Bob is 17 years old means that it is 17 years since his birth. That is, it is a description of time. So the statement is clear and complete.
Therefore, the correct choice is 3.
Answer : Ship A is 20 miles at an angle of 45
0 from ship B
Basic Concept of Vectors
A vector has both magnitude and direction. Let us take a very simple homework examples.
In diagram (a), just a line is shown. We don’t get any information from this diagram.
In diagram (b), a line segment is marked. The measure between the end points of the line segment gives a magnitude. So the measure of XY is a scalar.
In diagram (c), in addition to the magnitude of the line segment, an arrow mark is seen pointing the direction from X to Y. It means the line segment is directed from X to Y.
Now the line segment represents a vector and in vector notation it is denoted as vector XY
Types of Vectors
Below you could see different types of vectors
Zero Vector
A vector whose initial and terminal points coincide, is called a zero
vector (or null vector), and denoted as 0
Zero vector can not be assigned a definite direction as it has zero magnitude. Or, it may be considered as having any direction.
Unit Vector
A vector whose magnitude is unity (1 unit) is called a unit vector. The
unit vector in the direction of a given vector a is denoted by $\hat{a}$
Co-initial Vectors
Two or more vectors having the same initial point are called co-initial vectors.
Collinear Vectors
Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
Equal Vectors
Two vectors $\vec{a}$ and $\vec{b}$ are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as,
| a | = | b |
Negative of a Vector
A vector whose magnitude is the same as that of a given vector but direction is opposite to that of it, is called negative of the given vector. For example,vector XY is negative of the vector YX, and written as, XY = - YX
Components of a Vector
A vector can be resolved in horizontal and vertical directions. The components are expressed in terms of unit vectors î and ? in horizontal and vertical directions respectively.
The resolution of vectors help in operation of different vectors.
