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VECTORS
DEF: A vector is a quantity that has both magnitude and direction.
DEF: A scalar is a quantity that has magnitude but NO direction.
Ex. Vectors Ex. Scalars Force Temperature Velocity Time Displacement Mass Momentum Speed
Vector Notation
A – Boldface letters A
G
- Arrow above letter A – Magnitude of vector A
A vector is defined graphically by an arrow whose length is proportional to the magnitude of the vector quantity. The direction of the arrow points in the direction of the vector quantity.
Adding Vectors Graphically
Consider adding two vectors A and B graphically. The two vectors are shown below.
q q A
B
A
B
R
f
- Select an appropriate scale. (Ex. 20 cm = 5 N)
- Draw vector A to scale and in the proper direction.
- Draw vector B to the same scale with its tail at the tip of A and in the proper direction.
- The resultant vector R = A + B is the vector drawn from the tail of vector A to the tip of vector B.
- Calculate the magnitude of the resultant vector R using the selected scale and measure its direction with a protractor.
- This same process applies if you add more than two vectors.
This method of adding vectors graphically is also referred to as the head- to-tail method, analytical method, and geometric method.
Properties of Vectors
1. A + B = B + A (commutative Law)
- A + (B + C) = (A + B) + C (associative Law)
- A + (-A) = 0 (Negative of a vector)
- A – B = A + (-B) (vector subtraction)
- B = s A (vector multiplied by a scalar)
A
B
A
B
B
A
A + B
B + A (1)
A
B
C
B
C
(2)
(3) - A^ A = 0
A B
- B
A
B A - B (4)
Components of A Vector Consider a two dimensional vector A. The vector can be written in unit vector notation.
x
y
q Ax i
Ay j
A
A = A (^) x i + A (^) y j
A (^) x = A cos θ A (^) y = A sin θ
A (^) x and A (^) y are not vectors. They are scalar quantities that can be positive or negative depending on the angle θ. Ax and A (^) y are called the scalar components of vector A.
2 2 A = A = A x + Ay Magnitude of vector A
y x
A tan = A
θ
1 y x
A tan A
θ −^
= (^)
Direction of A relative to positive x-axis
Adding Vectors Using Component Method
Consider adding three 2-D vectors A , B , and C :
A = A (^) x i + A (^) y j B = B (^) x i + B (^) y j C = C (^) x i + C (^) y j
- Add the x-components and y-components of each vector to obtain the resultant vector R in unit vector notation.
R = A + B + C = (Ax i + A (^) y j) + ( B (^) x i + B (^) y j) + (Cx i + C (^) y j)
R = (A (^) x + B (^) x + Cx) i + (A (^) y + B (^) y + C (^) y) j
R x = Ax + Bx + Cx
R y = Ay + B y + Cy
R = R ix^ ˆ^ + R y ˆ j
G
- Calculate the magnitude of the resultant vector R.
2 2 R = R x + Ry
- Calculate the direction of R relative to the positive x – axis.
x
y
q Rx i
Ry j
R
y x
tan = R R θ
1 y x
tan R θ (^) R = −^ ^
- Same procedure applies if you add more than 3 vectors. However, if the vectors are 3D, then you must specify the direction of the resultant vector R relative to the positive x, y, and z axis.
Acceleration Vector
Again consider a particle moving between two points P and Q.
ri r (^) f
v (^) i
v (^) f
P
Q
Y
X
v (^) i
v (^) f
DV
ti
tf
v (^) i + ∆ v = v f
∆ v = v (^) f − v i Change in Velocity
f i ave f i
a t t t
∆^ − = = ∆ −
v v^ v Average Acceleration
Using calculus the result for the instantaneous acceleration is given by:
a = a x i + ay j + az k Instantaneous Acceleration
A particle will experience an acceleration whenever its velocity changes. This will occur when:
- the magnitude of v changes (linear motion)
- the direction of v changes (circular motion)
- both the direction and magnitude of v change
- (^) m^ V^ i^ m
V (^) f
V
V V
m
m
Vi
V (^) f
