2. SCALARS:
• A scalar is any positive or negative quantity that can be
completely specified by its magnitude for example length,
mass, time.
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3. VECTORS:
• A vector is any physical quantity that requires both a magnitude and a direction
for its complete description for example force.
• It is graphically shown by an arrow, the length of the arrow represents the
magnitude of the vector and the angle between the vector and a fixed axis
defines the direction of its line of action.
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4. MULTIPLICATION & DIVISION OF A VECTOR BY A
SCALAR
• If a vector is multiplied or divided by a positive scalar, its magnitude is increased
or decreased by that amount. If it is multiplied or divided by a negative scalar, it
will also change the direction sense of the vector.
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5. VECTOR ADDITION USING PARALLELOGRAM RULE
• First join the tails of the vectors to make them concurrent.
• From the head of B draw a line parallel to vector A. Draw another line from the
head of vector A parallel to vector B. Where the two lines intersect is point P.
• The diagonal of this parallelogram that extends to point P is the resultant R.
• R = A + B
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6. VECTOR ADDITION USING TRIANGLE RULE
• Vector addition is done using a triangle rule. Vector A is added to B using the
head to tail rule.
• Vector A is added to vector B by joining the head of vector A with the tail of
vector B.
• The resultant vector R extends from the tail of vector A to the head of vector B.
• R= A+B = B+ A
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7. VECTOR SUBTRACTION
• R = A – B
• Reverse the direction of vector B and then using the triangle rule we add the two
vectors as R = A – B = A + (-B)
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8. FORCES IN 2D – RESULTANT OF FORCES
• Finding Resultant of Forces:
• FR = F1 + F2 (Can be added using parallelogram rule (b) or triangular rule (c))
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9. ADDITION OF MORE THAN TWO FORCES
• If more than two forces are to be added, successive applications of parallelogram
rule can be applied. For example FR = F1 +F2 + F3 = (F1+F2) + F3
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10. COMPONENTS OF FORCES IN 2D
• Force F is to be resolved in two components along the axis u & v
• Using parallelogram, draw two lines from the tip of force F parallel to axis u and
axis v respectively.
• These parallel lines intersect with the axis u & v, and the components of forces
are established by joining the tail of the force F to these intersection points.
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12. PROPERTIES OF A PARALLELOGRAM
There are four important properties of parallelograms to
know:
• Opposite sides are congruent (AB = DC).
• Opposite angels are congruent (D = B).
• Consecutive angles are supplementary (A + D = 180°).
• If one angle is right, then all angles are right.
Note:
• Two line segments are congruent if they have the same
length.
• Two angles are congruent if they have the same measure.
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13. NUMERICAL EXAMPLE
• The screw eye is subjected to two forces F1 & F2. Determine the magnitude and
direction of the resultant force
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19. VECTORS IN 3D
• Right-Handed Coordinate System.
• Thumb point to the positive z axis, the fingers curl from positive x axis towards
positive y axis
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20. VECTORS IN 3D
• Rectangular components of
vectors:
• A = Ax + Ay + Az
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• Cartesian vector representation:
• A = Ax i + Ay j + Az k
22. UNIT VECTOR22
UA is a unit vector of A, in the direction of A.
Unit vector is the vector divided by its magnitude
23. POSITION VECTOR
• A position vector r is defined as a fixed vector which locates a point
in space relative to another point. For example if r extends from
origin to point P(x,y,z).
• r = xi + yj + zk
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24. • In more general case. the position vector may be directed from point A to point B
in pace.
24 POSITION VECTOR
25. POSITION VECTOR (CONTINUED)25
• Force F has the same direction and sense as the
position vector r directed from point A towards B.
This common direction is specified by unit vector u =
r/r
