How to solve vectors

Question

Introduction

Physical quantities are categorized into two primary types: scalar quantities and vector quantities.

Scalar Quantities: Scalar quantities are those quantity with just magnitude and does not specify its direction in space. Examples of scalars include length, mass, time, area, or volume. The description of a scalar quantity is complete once the unit is specified, as it is characterized only by its magnitude

Vector Quantities: Unlike scalars, a vector quantity is fully defined with both its magnitude (in unit) and its direction are specified. Examples of vector quantities include force, velocity, and acceleration. For vector quantities, both magnitude and direction are essential for a complete definition.

Instances of Scalars and Vectors

Examples of Scalars and Vectors

A distance of 5 meters is a scalar quantity, but

A displacement of 5 meters to the east is a vector quantity.

A temperature of 25&deg;C is a scalar quantity.

A gravitational acceleration of 9.8 m/s&sup2; acting downward is a vector quantity.

The force of gravity acting on a 7 kg object is a vector quantity.
	
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A bank balance of $1000 is a scalar quantity.
	
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A wind blowing at 15 km/h from the southwest is a vector quantity.

Vector representation

A vector quantity can be represented graphically by a line, drawn so that:

the length of the line denotes the magnitude of the quantity, according to some stated vector scale

the direction of the line denotes the direction in which the vector quantity acts.

The sense of the direction is indicated by an arrowhead.

The vector quantity AB is referred to as:

$\overline{AB}$

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The magnitude of the vector quantity is written as
|AB| or simply as AB.

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Note that in vectors; $\overline{AB}\ne \overline{BA}$

Rather, $\overline{AB}=-\overline{BA}$

The vector $\overline{BA}$ represents a vector of same magnitude as $\overline{AB}$  but opposite in direction.

Equality of Vectors

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

If a and B are both vectors, the for $\overline{AB}$ and $\overline{CD}$ to be equal, the both must have the same magnitude and direction as illustrated in the diagram.

If $\overline{AB}$  and $\overline{CD}$, then

AB = CD (magnitudes equal)

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Both vectors point in the same direction. This means they are parallel and have the same direction.

Similarly, if two vectors a and b are such that b = -a,

Then,

a and b have same magnitude.
	a and b have opposite directions.

Types of Vectors

a. Position Vector:
A position vector shows the location of a point in space with respect to a fixed origin or coordinate system.

b. Free Vector:
A free vector is defined only by its size (magnitude) and direction. It has no fixed starting point and can be represented by any arrow with the same length and direction, placed anywhere in space.

Most of the vectors we shall consider will be free vectors

Addition of vectors

The sum of two vectors, $\overline{AB}$ and $\overline{BC}$, is defined as the single or equivalent or resultant vector $\overline{AC}$

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c = a + b

To add two vectors a and b, place them head to tail&mdash;start vector b where vector a end. The resulting vector c is the one that goes from the start of vector a to the end of vector b. This vector c is the sum of a and b.

Example

if p is a force of 4 N, acting in the direction due west and, q is a force of 3 N, acting in the direction due south. Find the magnitude of the vector sum.

${{\text{r}}^{\text{2}}}\text{ =}{{\text{p}}^{\text{2}}}\text{+}{{\text{q}}^{\text{2}}}$

${{\text{r}}^{\text{2}}}\text{ =}{{\text{4}}^{\text{2}}}\text{+}{{\text{3}}^{\text{2}}}$

$=\sqrt{16+9}=\sqrt{25}=5$

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