2. In this lecture, you will learn Physical interpretations and
examples of scalars and vectors Basic operations on scalars and
vectors Properties of vector algebra How to represent vectors in
abstract space How to carry out the basic operations
mathematically
3. What are vectors? Vector: Quantity with magnitude and
direction Velocity Force Acceleration Momentum Torque v F a p
4. What are scalars? Scalar: Quantity with magnitude, but no
direction Temperature Pressure Time Energy Mass T P t m E
5. A joke What do you get when you cross a disease-carrying
mosquito with a mountain climber? ?
6. A joke Undefined! You cannot cross a vector with a
scalar.
7. What was that for? Disease-carrying mosquito = vector From
Latin vectus, to carry Vectors can be represented as arrows Can be
thought of as conveyor belts that carry objects from their tip to
their tail
8. What was that for? Mountain climber = scaler Pronounced the
same way as scalar Scalars are just real numbers (in physics,
usually with units) Their purpose is to scale vectors v 2v 3v
9. Vectors The precise definition of a vector is more than just
a quantity with magnitude and direction... ...but you can think of
vectors that way for now.
10. Geometric Representation A vector can be visualized as an
arrow. Direction the arrow points: direction of vector Length of
arrow: magnitude of vector v |v|
11. Vector Addition Finding A + B is not as simple as 1 + 1 =
2. Copy B starting from the tip of A. OR copy A starting from the
tip of B. A B A B Resultant vectorResultant vector AA ++
BBResultant vectorResultant vector BB ++ AA Notice that vector
addition is commutative, i.e., A + B = B + A. This is sometimes
called the parallelogram law.
12. Scalar Multiplication This one's as simple as 1 2 = 2. The
resultant vector has the same direction, but its length (magnitude)
is scaled by the scalar. If the scalar is negative, then the
resultant points in the opposite direction. v 2v 3v 1 2 v 1 2 v
2v
13. The Zero Vector Represented by bold 0 or by a zero with an
arrow on top Additive identity of vectors: A + 0 = A. Scaling a
vector by the scalar 0 gives the zero vector, i.e., 0A = 0. 0 A 0+
A=A 0 A=0
14. The Negative of a Vector Same magnitude, pointing in the
opposite direction The negative of A is written as -A. Also known
as additive inverse. Same as scaling by -1, i.e., (-1)A = -A.
Additive inverse property: A + (-A) = 0. A A A-A 0
15. Vector Subtraction Just like with numbers, subtraction is
defined as the addition of the additive inverse. That is, A B = A +
(-B). A B BAB
16. More Properties of Vector Algebra Add. associativity:
A+(B+C) = (A+B)+C Distributivity of multiplication over vector
addition: s(B + C) = sB + sC over scalar addition: (s + t)A = sA +
tA Mult. associativity: s(tA)=(st)A=t(sA) A B C A+B+CA+B B+C A 2A B
2B A+B 2(A+B) =2A+2B A 2A 3A 2A 2A+3A = (2+3)A = 5A A 2A 3A 2(3A) =
3(2A) = 6A
17. Magnitude of a Vector Denoted by |A|, same as the absolute
value symbol Can be thought of as the length of the arrow Some
readily seen properties: |-A| = |A| |sA| = |s||A| v |v| |-v| = |v|
-v 1 4 v=1 4 v
18. Vectors in a Coordinate System Drawn starting from the
origin Expressed in terms of their components x-comp = 2 y-comp = 3
x-comp = -1 y-comp = -6 x y
19. Unit Vectors Unit vector: any vector with magnitude 1 In
3-D space, it is convenient to define unit vectors whose directions
are along the +x, +y, and +z axes. x y z i j k
20. Algebraic Representation of Vectors The standard unit
vectors i, j, and k form what is called an orthonormal basis of 3-D
space, because any 3-D vector can be expressed uniquely in terms of
i, j, and k. A 3-D vector denoted by can be written as xi + yj +
zk. x y z r = xi + yj + zk xy z
21. Algebraic Addition of Vectors Just add the vectors
component by component. A = 2i + 3j x y B = 3i 7j A + B = (2+3)i +
(37)j A + B = 5i 4j x-components y-components
22. Algebraic Multiplication of a Vector by a Scalar Multiply
each component by the scalar. x A = -2i + j 3A = -6i + 3j
24. Example 2 If P = 2j + 7k and R = 3i 4j 2k, find Q such that
4P + 3Q = R. Solution: Start with 4P + 3Q = R. Isolate Q. Q =
(1/3)R (4/3)P Q = (1/3)(3i 4j 2k) (4/3)(2j + 7k) Q = i (4/3)j
(2/3)k (8/3)j (28/3)k Q = i 4j 10k
25. A few words of caution You cannot add a scalar and a
vector. For example, 4 + 4j is undefined. You cannot multiply two
vectors. For example, (4i j)(3j + k) is undefined. However, in the
next lecture, you'll learn about two different kinds of product
operations you can do on vectors: the dot product and the cross
product.
26. Problems 1.If A = 3j + 6k and B = -2i 2j + 5k, find: (a)2A
3B (b)-4A + B 2.If C = 2i + 5j and D = -4i + j, find E such that C
2E = 3D. 3.Given A = 2j + 3k, B = 3i 6j + 4k, C = 26i + 9j 6k, and
r = aA + bB + cC, find a, b, and c if (a) r = 18i 38j + 21k (b) r =
-32i + 19j + 22k
