Vectors and Scalars
• A Scalar is a physical quantity with magnitude (and units). Examples: Temperature, Pressure, Distance, Speed
• A Vector is a physical quantity with magnitude and direction: Displacement: Washington D.C. is ~ 180 miles N of
Newport News Wind Velocity: 20mi/hr towards SW
Components of a vector, as an alternate to magnitude and direction
• 100 miles 30 degrees north of east, is equivalent to 86.6 miles east followed by 50 miles north
86.6mi
50mi100mi
N
E
Labels for Components of a Vector
• To free ourselves from the points of the compass, we will use x & y instead of E & N
• Vector , magnitude• Components (as ordered pair)
A
AA
),( yx AAA
),(),( yxrrr yx
Trigonometry and Vector Components
• Trigonometry is not a pre-requisite for this course.• Today you will learn ½ of trigonometry, and all
that you need for this course.• In this discussion, we always define the direction
of a vector in terms of an angle counter-clockwise from the + x-axis.
• Negative angles are measured clockwise.
Trigonometry and Circles
• The point P1=(x1,y1) lies on a circle of radius r.
• The line from the origin to P1 makes an angle w.r.t. the x-axis.
• The trigonometric functions sine and cosine are defined by the x- and y-components of P1:
x1
y1r
P1
45-45-90 triangle
• By symmetry, x= y
• Pythagorean Theorem: x
+ y
2 = r2
2· x = r2
x= r/2
• cos(45º) = x /r • cos(45º) = 0.7071• sin(45º) = 1/ 2
Vector Addition:Graphical(use bold face for vector symbol)
• A, B, and C are three displacement vectors. Any point can be the origin for a
displacement
• The vector B = 3 paces to E. Notice that B has been translated
from the origin until the tail of B is at the head of A.
• This is the “head-to-tail” method of vector addition.
• Vector addition is commutative, just like ordinary addition: D = A+B+C = C+B+A
Vector Addition, Components
• When we add two vectors, the components add separately: Cx = Ax + Bx
Cy = Ay + By
Velocity Vectors
• Each fish in a school has its own velocity vector.
• If the fish are swimming in unison, the velocity vectors are all (nearly) identical
• We draw each vector at the position of the fish.
Scalar MultiplicationMultiplying a vector by a scalar
• Multiplying a vector by a positive scalar quantity simply re-scales the length (and maybe units) of the vector, without changing direction.
• Multiplying a vector by a negative number reverses the direction of the vector.
x
yA
A
))(6.0( A
Vector Subtraction
• Subtraction is just addition of the additive inverse
1212 vvvv
2112
2112 get to toadd vector to
vvvv
vvvv
x
y
2v
1v
1v
12 vv
Average Velocity Vector
• Net displacement (vector) multiplied by reciprocal of elapsed time (scalar)
1212
12
12
1 )(
ttrr
t
r
tt
rrvav
r1
r2
A whale comes to the surface to breathe, and then dives at an angle 20.0° below the horizontal. Answer the following questions if the whale continues in a straight line for 140 m. (a) How deep is it? (b) How far has it traveled horizontally?
Example 1
Example 2
Consider the vectors in Figure 3-36, in which the magnitudes of A, B, C, and D are respectively given by 15 m, 20 m, 10 m, and 15 m. Express the sum, A + C + D, in unit vector notation.