x̄ - > Phy100 Vector Scalar and Vector Quantities

 CHAPTER 1: Vector Scalar and Vector Quantities

Many quantities (e.g. volume) have no direction associated with them. These quantities which normally have magnitude only are called scalar quantities. Scalar quantities include, mass, time, density, work, temperature, amount of money etc. There are other quantities, which have both magnitude and direction. These are vector quantities. These include displacement, velocity, force, acceleration, and electric field etc. A vector quantity is represented by an arrow drawn to scale. The length of the arrow represents the magnitude and the direction of the arrow represents the direction of that vector.Vector Addition

Vectors are added using the geometric method. Vectors don’t obey ordinary rules of algebra. They combine according to certain rules of addition and multiplication. Vectors are added by geometrically connecting the head to the tail of the other vector and drawing a straight line between the other tail and head of the vectors. This gives you a resultant vector

(There is a diagram of vectors A, B, C, and P)AB + BC = AC — Resultant vector

The resultant of a number of force vectors is that single vector which would have the same effect as all the original vectors together.Commutative Law of Vector Addition

AB + BC = BC + AB. During vector addition it does not matter with the vector you begin with first. The resultant or effective vector will be the same.Associative Law of Vector Addition

Consider more than two vector which are to be added together. Draw to scale each vector in turn, taking them in any order of succession. The tail end of each vector is attached to the arrow end of the preceding one. The line drawn to complete the polygon is equal in magnitude to the resultant of equilibrant. An equilibrant of a number of vectors is that vector which would balance all original vectors taken, together. It is equal in magnitude but opposite in direction to the resultant.

For associative law of vector addition  

AB + BC + CD = AD, ⇒ (AB + BC) + CD = AB + (BC + CD). Hence the ordering of the vectors makes no difference as far as their addition is concerned. This is the associative law of vector addition.

Magnitude of Resultant Vector and Angles between the Vectors

Consider the figure given

 below where AB = v₁, BC = v₂, BD = v₃, DC = v₄, Sinθ, Angles CBD = α, CAB = α, ACB = β.  

AB + BC = AC, v₁ + v₂ = v. To compute the magnitude of v we have (AC)² = (AD)² + (DC)².  

But AD = AB + BD = v₁ + v₃, Cos θ, DC = v₂ Sinθ. Therefore (AC)² = v₁² + v₂ Cosθ)² + (v₂ Sinθ)² = v₁² + v₂² + 2 v₁ v₂ Cosθ or (v₁² + v₂² + 2 v₁ v₂ Cosθ)¹/². To determine the angle we need only find angle α. From the figure we see that in triangle ACD, CD = AC Sinθ, and in triangle BDC, BC sin α = AC Sin α = BC Sinθ.  

Similarly, BE = v₁ Sin α = v₂ Sin β. When we combine we get,

When v₁ & v₂ are perpendicular θ = π/2, and from v = (v₁² + v₂² + 2 v₁ v₂ Cosθ) we have v = (v₁² + v₂²)¹/² and tanα = (Opposite)/(Adjacent) = v₂/v₁.

Subtraction of Vectors

The negative of a vector is another vector of equal magnitude but opposite direction: e.g.  

(Diagram of two vectors pointing in opposite directions)