SS1 Second Term Further Mathematics Lesson Note – Vectors I

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WEEK 4

TOPIC: VECTORS 1

CONTENT:

• Scalars and Vectors
• Types of vectors.
• Components of vectors
• Algebra of vectors
• The triangle law, parallelogram law and Resolution of vectors
• Position vectors

SUB – TOPIC 1: Scalars and Vectors/Types of vectors

Scalars and Vectors

There are two kinds of physical quantities

1. Those which have only magnitude
2. Those which have both magnitude and direction

When we talk of the temperature of a patient as exceeding the normal body temperature of 98.4⁰F or the mass of a luggage as 34kg, or the density of dry air at STP as 1.293kg.m^-3 etc, we are referring to some specific magnitude in each case and these quantities go by the name scalars.

Definition: A scalar is that quantity which possesses only magnitude

Besides temperature, mass and density, other examples of scalars are; energy, speed, length and time.

Suppose an engine boat moves with a velocity of 10knots, 45⁰ East of North, then the displacement of the boat can be found at any given time if we know the starting point. We have two types of displacement namely;

Localized or free displacement: for example, a displacement 40km east is the same in Lagos as in Accra.

Line located displacement; as in forces. When a force acts on a body, the body tends to move in the direction of the line of force.

All quantities that behave or have the same characteristics as displacement are called Vectors.

Definition:

A vector quantity is that which has magnitude, direction and sense. The following are vector quantities; force, momentum, velocity, acceleration, displacement, electric field strength and magnetic induction. Vectors which have no particular directions associated with them are called Free Vectors.                                                                                                                          A vector which is located along a straight line is called a Line located vector. Force is an example of a line located vector. When a vector is applicable to a particular point, it is called Point located vector.

In printed work, vectors will be shown in print to distinguish them from scalars that are in light print.

The length or magnitude of a vector a is denoted lal. The magnitude of a vector is sometimes called its Modulus or Absolute value or Numerical value.

Vector which has a magnitude of zero and no particular direction is called a Zero vector or Null vector and is denoted by 0.

A vector which has an absolute value of unity is called a Unit vector.

The vector which has the same absolute value as a vector a but a direction opposite to the vector a is called the Negative vector of a and is denoted –a.

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