Vectors and Scalars

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Physical quantities are divided into two categories:

  1. Scalars quantities: They have only the magnitude. For example, mass, length, time, temperature etc. The scalars quantities can be added, subtracted, multiplied and divided just as the ordinary numbers.
  2. Vectors quantities: They have both magnitude and direction. for example velocity, force, electric field, torque etc.

A vector can be defined as a line segment having a specific direction and a specific length. The length is the magnitude of the vector and with an arrow indicating the direction. The direction is from the tail of the vector to its head.

Figure:5.a

A vector is denoted by the letter in bold (for example vector p is written as p) or it can be represented by an arrow placed over a letter (for example vector is written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>→</mo></mover></math>.

Position and displacement vectors:

A position vector describes the position of a point in a coordinate system.

Figure:5.b

If P is the position of an object at time t, then OP is the position vector of the object at that time. It is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>r</mi><mo>→</mo></mover></math>.

The displacement vector describes the position of a point with reference to a point other than the origin of the coordinate system.

Figure:5.c

An object moves from a point P to point Q. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>r</mi><mn>1</mn></msub><mo>→</mo></mover></math>  is the position vector at a time $t_{1}$ and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>r</mi><mn>2</mn></msub><mo>→</mo></mover></math> is the position vector at a time $t_{2}$.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mover><mi>r</mi><mo>→</mo></mover></math> is the displacement vector directed from point P to Q, corresponding to the motion of the object from P to Q.

Parallel vectors and antiparallel vectors:

Parallel vectors act along the same direction and they are collinear. The angle between them is $0^{0}$.

Figure:5.d

Antiparallel vectors act in the opposite direction and they are collinear. The angle between them is $180^{0}.$

Figure:5.e

Equality of vectors:

Two vectors are said to be equal if they have the same magnitude and direction.

Figure:5.f

Unit vector:

Unit vector has a unit magnitude and points in a particular direction. Any vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>→</mo></mover></math>
can be written as the product of unit vector in that direction and magnitude of the given vector.

Unit vector is expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>^</mo></mover></math>.

We can write, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>→</mo></mover></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>p</mi><mo>^</mo></mover></math> $p$

A unit vector has no dimensions. Unit vectors along the positive X, Y and Z -axes of the coordinate system can be expressed as
<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>i</mi><mo>^</mo></mover><mo>,</mo><mover><mi>j</mi><mo>^</mo></mover><mo> </mo><mi>a</mi><mi>n</mi><mi>d</mi><mo> </mo><mover><mi>k</mi><mo>^</mo></mover></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mover><mi>i</mi><mo>^</mo></mover></mfenced><mo>=</mo><mfenced open="|" close="|"><mover><mi>j</mi><mo>^</mo></mover></mfenced><mo>=</mo><mfenced open="|" close="|"><mover><mi>k</mi><mo>^</mo></mover></mfenced><mo>=</mo><mn>1</mn></math>.

Figure:5.g

 

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