Scalars and Vectors. A scalar quantity is one that can be described by a single number: temperature, speed, mass. A vector quantity deals inherently with both magnitude and direction : velocity, force, displacement, acceleration. Scalars and Vectors.
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A scalar quantity is one that can be described
by a single number:
temperature, speed, mass
A vector quantity deals inherently with both
magnitude and direction:
velocity, force, displacement, acceleration
Arrows are used to represent vectors. The
direction of the arrow gives the direction of
the vector.
By convention, the length of a vector
arrow is proportional to the magnitude
of the vector.
8 lb
4 lb
The plane travels with a velocity relative to the ground which is vector sum of the plane’s velocity (relative to the air) plus the wind velocity
The resultant is a combination of both motions
These are the rules you must play by to add vectors and find the resultant:
Draw each vector to scale and label
Vectors should be drawn in order as they occurred and labeled (I.e.v1, v2…..)
Draw each successive vector starting from the tip of the preceding vector. (This is called the “tip to tail” method
The vector sum is found by drawing the resultant vector arrow from the tail of the first vector to the tip of the last vector
For twovector operations in two dimensions the resultant forms the diagonal of a parallelogram with the component vectors forming the sides.
v1
Vector v1 is 1 unit east or +1 unit
A1
v2
A2
Vector v2 is 3 units west or  3 units
Vector (v1 + v2) is 2 units west or  2 units
R = A1 + A2
v1 + v2 = R
v1
v2
R = Runits @ 0 ( measured from East counterclockwise)
Use a ruler and vector rules to find the resultant of the following vector additions. Record your answer using vector convention.
R = F1+ F2
F1
F2
F1
F2
P1
P1
P2
R = P1+ P2
P2
R = P2+ P1
P1
When a vector is multiplied
by 1, the magnitude of the
vector remains the same, but
the direction of the vector is
reversed.
Finding the components of a vector
To find the magnitude of a vector quantity in a particular direction you can use the parallelogram rule
Draw the vector and treat it as the resultant
Form a parallelogram around the resultant in the two directions of interest. In physics problems these are usually perpendicular directions to simplify the problem
Draw the component vectors along the sides of the parallelogram
ah
Pythagorean Theorem:
a2 = ah2 + av2
av
a
a
Eparalel
E
E
Pythagorean Theorem:
E2 = Eparalel2 + Eperpendicular2
Eperpendicular
Using Trig to solve 2DVector Problems
RightAngled Trig Relationships
Trig functions for rightangled triangles are just geometric ratios. You probably memorized these ratios in your math classes using So/h..Ca/h..To/a. Here they are once more!
sin = o / h
cos = a / h
h
o
tan = sin / cos = o / a
a
Follow these rules!
Sketch vectors on a coordinate system. (Two perpendicular axes)
Find (resolve) the components of all vectors in the two directions of interest using the parallelogram rule. Make sure that you use the normal conventions of + and – arithmetic quantities.
Sum the components in the two directions of interest.
Find the magnitude of R using Pythagorean theorem
Use the tangent rule to determine the direction R as measured from the East by convention or a specified direction
Use trig relationships to determine i) the other component and ii) the resultant R
i) tan = vv / vh
R
vv
vv = vhtan
vv = (35 m/s) tan 600
600
vh= 35 m/s
vv = 60.6 m/s
ii) R = (vv2 + vh2)
R = ((60.6 m/s)2 + (35 m/s)2)
R = 70 m/s
= 1800  600 = 1200
R = 70 m/s @ 1200
Determine the resultant R of the following vectors by using the horizontal and vertical components of A and B
R = 100 Newtons
A cos 600
A sin 600
A = 80 Newtons
+
80 N cos 600
+80 N sin 600
B= 60 Newtons
 40 N
+ 69.3 N
Av
Bv
B cos 300
B sin 300
600
300

+
60 N sin 300
+60 N cos 300
AH
BH
R = (RH2 + Rv2)
+ 52 N
+ 30 N
R = ((12 N)2 + (99.3 N)2)
+ 12 N
+ 99.3 N
R = 100 N
= tan1 (Rv / RH)
= tan1 (99.3 N / 12 N)
R = 100 N @ 830
= 830