Motion in a Plane

Scalars and Vectors

• A scalar quantity is a quantity with magnitude only.
• It is specified completely by a single number, along with the proper unit.
• Examples : the distance between two points, mass of an object, the temperature of a body……etc
• Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers.

• A vector quantity is a quantity that has both a magnitude and a direction and obeys the triangle law of addition or equivalently the parallelogram law of addition.
• Examples : displacement, velocity, acceleration and force…….etc.
• The magnitude of a vector is often called its absolute value.

Position and Displacement Vectors

Consider a body moving in X-Y plane.

$\Rightarrow \mathrm{Here} \overrightarrow{\mathrm{OP}} \mathrm{and} \overrightarrow{\mathrm{OQ}} \mathrm{are} \mathrm{the} \mathrm{position} \mathrm{vectors} \mathrm{of} \mathrm{P} \mathrm{and} \mathrm{Q}$.

$\Rightarrow \mathrm{Here} \overrightarrow{\mathrm{PQ}} \mathrm{is} \mathrm{the} \mathrm{displacement} \mathrm{vector}.$

Equality of Vectors

• Two vectors A and B are said to be equal if, and only if, they have the same magnitude and the same direction.

Multiplication of Vectors by Real numbers

• Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A :
  |λ A| =λ | A| if λ > 0.
• Multiplying a vector A by a negative number −λ gives another vector whose direction is opposite to the direction of A and whose magnitude is λ times |A|.

Addition of two Vectors (Graphical Method)

• Two vectors can be added, if both of them are of same nature.
• Types of vector addition are:

1. Triangle Law of Vector Addition.

2. Parallelogram Law of Vector Addition.

3. Polygon Law of Vector Addition.

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