PHY1012F VECTORS. Gregor Leigh gregor.leigh@uct.ac.za. VECTORS. Resolve vectors into components and reassemble components into a single vector with magnitude and direction. Make use of unit vectors for specifying direction.
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PHY1012FVECTORS
Gregor Leighgregor.leigh@uct.ac.za
VECTORS
Learning outcomes:At the end of this chapter you should be able to…
– A physical quantity with magnitude (size) but no associated direction.E.g. temperature, energy, mass.
Vector
– A physical quantity which has both magnitude AND direction.E.g. displacement, velocity, force.
Algebraically, we shall distinguish a vector from a scalar by using an arrow over the letter, .
The important information is in the direction and length of the ray – we can shift it around if we do not change these.
Note:r is a scalar quantity representing the magnitude of vector , and can never be negative.
N
20 km
60°
10°
15 km
= 74°
Notes:
(The (entire) system can be rotated – any which way – to suit the situation.)
“Running the movie backwards” resolves a single vector into two (or more!) components.
“Running the movie backwards” resolves a single vector into two (or more!) components.
“Running the movie backwards” resolves a single vector into two (or more!) components.
??!
Even if the number of components is restricted, there is still an infinite number of pairs into which a particular vector may be decomposed.
Unless…
is now constrained to resolve into and , at right angles to each other.
Note that, provided that we adhere to the right-handed Cartesian convention, the axes may be orientated in any way which suits a given situation.
Ax, the scalar component of (or, as before, simply its component) along the x-axis …
8
6
4
2
0
0
2
4
6
8
Ax = +6 m
Ay = +3 m
8
6
Ax = +6 m
Ay = +3 m
4
2
-8
-6
-4
-2
0
-8
-6
-4
-2
-2
Ax = +6 m
Ay = +3 m
-4
-6
-8
4
2
Ax = –2 m
Ay = +4 m
-4
-2
2
-2
4
2
Ax = –6 m
Ay = –5 m
-8
-4
4
-2
The components of are…
PHY1012F
4
2
Ax = –6 m
Ay = –5 m
–8 m
–3 m
4
-4
-8
-4
20
PHY1010W
Ax = +Acos m
Ay = –Asin m
PHY1010W
Ax = –Asin m
Ay = –Acos m
Note that we can (re)combine components into a single vector, i.e. (re)write it in polar notation, by calculating its magnitude and direction using Pythagoras and trigonometry:
(On this slide!)
1
1
Given a 12 m/s velocity vector which makes an angle of 60° with the negative x-axis, write the vector in terms of components and unit vectors.
vy = +vsin60°
v = 12 m/s
vx = –(12 m/s)cos60° vx= –6.00 m/s
vx = –vcos60°
vy = +(12 m/s)sin60° vy= +10.4 m/s
Hence:
and
ThusDx= Ax + Bx + Cx andDy= Ay + By + Cy
In other words, we can add vectors by adding their components, axis by axis, to determine a single resultant component in each direction. These resultants can then be combined, or simply presented in unit vector notation.
By
Dy = Ay + By
Ay
=
+
Ax
Bx
Dx = Ax + Bx
Dy = Ay + By
its polar form is easily reconstituted from Dx and Dy
using and
Dx = Ax + Bx
N
20 km
60°
10°
15 km
= 74°
N
20 km
+15 sin10°
+2.60
+15 cos10°
+14.77
10°
–18.79
–20 cos20°
–20 sin20°
–6.84
15 km
–16.2
+7.93
20°
N
150 m at 45° west of south. After a fourth unmeasured displacement he finds himself back where he started. Determine the magnitude and direction of his fourth displacement.
30°
50 m
100 m
150 m
45°
N
Rx = 100 – 25 –106 + Dx = 0
Dx = 31
30°
Ry = 0 +43.3 –106 + Dy = 0
Dy = 62.7
50 m
100 m
150 m
45°
100
0
–25
43.3
–106
–106
63.7
69.9
?
?
?
31
62.7
?
VECTORS
Learning outcomes:At the end of this chapter you should be able to…