VECTORS & TRIANGLES
VECTORS�(review)
A VECTOR quantity
is any quantity in
physics that has
BOTH MAGNITUDE
and DIRECTION
Vector Example | Magnitude and Direction |
Velocity | 35 m/s, North |
Acceleration | 10 m/s2, South |
Force | 20 N, East |
An arrow above the symbol
illustrates a vector quantity.
It indicates MAGNITUDE and
DIRECTION
VECTOR APPLICATION�(review)
ADDITION: When two (2) vectors point in the SAME direction, simply add them together
EXAMPLE: A man walks 46.5 m east, then another 20 m east. Calculate his displacement relative to where he started.
66.5 m, E
MAGNITUDE relates to the size of the arrow and
DIRECTION relates to the way the arrow is drawn
46.5 m, E
+
20 m, E
VECTOR APPLICATION�(review)
SUBTRACTION: When two (2) vectors point in the OPPOSITE direction, simply subtract them
EXAMPLE: A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started.
26.5 m, E
46.5 m, E
-
20 m, W
NON-COLLINEAR VECTORS
When two vectors are PERPENDICULAR to each other, you must use the PYTHAGOREAN THEOREM
Example: A man travels 120 km east then 160 km north. Calculate his resultant displacement.
120 km, E
160 km, N
the hypotenuse is
called the RESULTANT
VECTOR X = Vx
VECTOR Y = Vy
START
FINISH
WHAT ABOUT DIRECTION?
In the example, DISPLACEMENT is asked for and since it is a VECTOR quantity, we need to report its direction.
N
S
E
W
N of E
E of N
S of W
W of S
N of W
W of N
S of E
E of S
NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components TIP TO TOE.
N of E
TRIGONOMETRIC FUNCTIONS
NEED A VALUE – ANGLE!
Just putting N of E is not good enough (how far north of east?). We need to find a numeric value for the direction.
N of E
160 km, N
120 km, E
To find the value of the angle we use a Trig function called TANGENT.
θ
200 km
So the COMPLETE final answer is : 200 km, 53.1°North of East (or NE)
What are your missing components?
Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal (Vx) and vertical components (Vy)?
65 m
25º
HC = ?
VC = ?
The goal: ALWAYS MAKE A RIGHT TRIANGLE!
To solve for components, we often use the trig functions since and cosine.
Vy
Vx
From home a car drives 15 m north and 8 m west. Calculate the car's resultant displacement with respect to due north.
Example 1
Example 1
From home a car drives 15 m north and 8 m west. Calculate the car's resultant displacement with respect to due north.
15 m N
8 m W
Rv
θ
The Final Answer : 17 m @ 28.1°West of North (or NW)
A jogger runs 30 m at 32°South of East. Calculate the runner’s horizontal component.
Example 2
Example 2
A jogger runs 30 m at 32°South of East. Calculate the runner’s horizontal component (Vx).
cosθ = adjacent
hypotenuse
adj = (hyp)(cosθ)
adj = Vx = 30cos32
Horizontal Component Vx = 25.4 m
30 m
32°
H.C. =?
A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.
Example 3
Example 3
A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.
35 m, E
20 m, N
12 m, W
6 m, S
-
=
23 m, E
-
=
14 m, N
23 m, E
14 m, N
The Final Answer: 26.93 m, 31.3 degrees North of East (or NE)
R
θ