P.1
PHYSICS
Vectors
1.1Introduction
a)definition
-scalar quantities can be completely specified by a magnitude. They can be added algebraically.
-vector quantities cab be specified by both a magnitude and a direction.
They can be added geometrically or analytically.
b)symbols of vector quantities
-Vectors quantities are usually represented by boldface type
e.g. V in printed material.
When written by hand, the designation is often used.
c)representation of vector
-A vector is a line drawn to represent a vector quantity in
(i)direction, by its arrow head, and
(ii)magnitude, by its length.
e.g. consider 2 vectors and,
we have = - 4
d)example in mechanics
scalar quantities / vector quantitieslength / l / displacement /
mass / m / force /
time / t / velocity /
volume / V / acceleration /
speed / v / momentum /
energy / W/E / torque / couple /
pressure / P / angular momentum /
density / q
temperature / θ/T
potential difference (voltage) / V
e)The resultant and equilibrant
-the resultant of a number of similar vectors is that single vector which would have the same effect as all the original vectors taken together.
-the equilibrant of a number of vectors is the single vector which is equal in magnitude and oppositely directed to their resultant.
f)Zero vector and until vectors
-the null vector (or zero vector) is a vector with zero magnitude and no specific direction.
-a unit vector is a vector whose magnitude is 1.
The basic unit vectors in rectangular coordinates are
= unit vector in positive x direction
= unit vector in positive y direction
= unit vector in positive z direction
1.2Addition of vectors
a)Parallelogram method (for adding 2 vectors)
-the resultant of 2 vectors acting at any angle may be represented by the diagonal of a parallelogram.
-the resultant vector can be measured graphically or calculated by cosine rule.
b)Polygon method (adding any number of coplanar vectors geometrically)
-Start at any convenient point and draw (in scale) each vector arrow in turns. They may be taken in any order of succession. The tail end of each arrow is attached to the tip end of the preceding one.
c)Rectangular method (for both 2-dimensional and 3-dimensional cases)
-Resolution and composition of vectors are usually resolved into vertical and horizontal components. Then the resultant vector is the composite of all the components together.
e.g. Find the resultant of the four coplanar forces act on a body at point 0 as shown below.
solution:
vector / x-component / y-component80 / 80 / 0
100 / 100 cos 45º / 100 sin 45º
110 / -110 cos 30º / 110 sin 30º
+) / 160 / -160 cos 20º / -160 sin 20º
/ Rx = 94 / Ry = 71
Since = Rx + Ry
i.e. = -94 + 71
R = = = 118N
tanθ= Ry / Rx
θ= -37ºor 143º(0ºθ 360º)
θ= 143º only (It is because R lies on the second quadrant)
e.g. Find the magnitude and direction of the vector d.
Given that
solution:
vector / x-component / y-component/ 3 / -4
/ -1 / 0
+) / / 0 / 8
/ dx = 2 / dy = 4
hence, = 2 + 4
d = = =
tanθ= dy / dx = = 2
θ= 63.4ºor 243.4º
θ= 13.4º only (it is because d lies on first quadrant)
d)Properties of vector addition
(i) (commutative law)
(ii) (commutative law)
1.3Subtraction of vectors
a)Parallelogram method
Since
summary:
b)Application on relative velocity
IfA= velocity of A
B= velocity of B
AB= velocity of B w.r.t. A
then AB = B - A
e.g.Two oars start at the same point with uniform velocities.
Car A moves to the north with speed 10m/s while car B moves to the east with speed 4 ms-1. Find the velocity of car A observed by a passenger in car B.
Solution:
BA = A = B = -4 + 10
VBA = =
tanθ=
θ= ______or ______
θ= ______only (it is because VBA lies on the second quadrant)
e.g.Find the relative velocity of B w.r.t. A
AB = B –A
VAB =
1.4Multiplication of vectors
a)Product of a scalar k and a vector
K () = k
-It is defined as a new vector whose magnitude is k times the magnitude of . The new vector has the same direction as a if k is positive and opposite in direction if k is negative.
b)Scalar (dot) product of 2 vectors and
(i)definition – this product id a scalar and is equal to the product of the magnitude of one vector and the component of the other vector in the direction of the first.
(ii)properties
(1)
(2)
(iii)dot product of rectangular unit vectors
(1)
(2)
e)Vector (cross) product of 2 vectors a and b
(i)
definition – this product is a vector. Its magnitude is equal to the product of the magnitude of one vector and the component of the other vector in the direction perpendicular to the first.
Its direction is perpendicular to the plane containing vectors , and is determined by right hand screw rule.
(ii)Properties
(1)
(2)
(iii)Cross product of rectangular unit vectors
(1)
(2) ; ;
d)Examples of multiplication of vector qualities
(i)dot product e.g. work done on a body by a force F after a displacement s.
W = = s cosθ (θis the angle between and )
frictionlessfrictionless
surfacesurface
W= (10) (5) cos 0˚W= (10) (5) cos 60˚
= 50J= 25J
(ii)cross product
e.g. magnetic force acting on a current carrying wire,
= magnetic field strength
= current
= length of the wire (in the direction of the current)
= magnetic force induced on the wire
Since = I x ,
is directed perpendicularly into this page which is consistent with the result obtained by using Fleming’s left hand rule.