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Chapter2. Vectors

2.1 Introduction: There are some physical quantities, which are completely described by a single number with a unit, for example; mass of a body. On the other hand, there are other quantities that need a direction specified along with the magnitude for a complete description.

The quantities which are described only via magnitudes are called Scalar quantities. For example if we say that we drive 10 miles, we are talking about the total distance traveled. Here we are talking about magnitude only (distance) i.e. a Scalar quantity.

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Those quantities which are fully described by magnitude as well as direction are known as Vector quantities. For example, traveling with a velocity of 20 miles per hour due south is not the same as traveling with a velocity of 40 miles per hour due east. These quantities are Vectors and it is important to distinguish them from scalars (physical quantities described by a single number). Examples of vector quantities are forces, velocity or the position of a robot etc.

Variables that are vectors will be indicated with a boldface variable, although it is common to see vectors denoted with small arrows above the variable.

2.1.1 Unit Vector:

A unit vector is a vector that has a magnitude of one. A vector representing a unit vector is also boldface, and it will have a carat (^) above it to indicate the unit nature of the variable. The unit vector x, when written with a carat, is read as "x-hat" as the carat looks like a hat on the variable.

2.1.2 Zero Vector:

The zero vector, or null vector, is a vector with a magnitude of zero. It is written as 0.

2.1.3 Parallel Vectors:

Vectors are parallel if they have the same direction. Both components of one vector must be in the same ratio to the corresponding components of the parallel vector.

2.1.4 Collinear Vectors:

Two vectors that are parallel to each other are called ``collinear'', as they can be drawn onto the same line.

2.1.5 Anti-parallel vectors: When two vectors V1 and V2 are in opposite directions, their magnitudes being same or not, we say that they are anti-parallel. If they have the same magnitude, the relation between the two vectors is

V1 = −V2 or

V2 = −V1

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