VECTORS

SCALARS

• A real number, hence maybe positive, negative or zero• Specified as a single number (with or without unit of

measure)• Example: time, mass, density, temperature• Combined using algebra

VECTORS

• Specified by giving its magnitude and direction• Example: force, displacement, velocity, acceleration,

momentum• Graphically represented by a directed line segment• magnitude indicated by the length of the arrow, • direction by the sense of the arrow head, measured as the

angle the line makes with the +x-axis• Maybe a radius vector or displacement vector• Symbolic represented by a bold-faced letter, or a letter capped

with an arrow or half arrow

,x y A 2 1 2 1,x x y y A

• Equal vectors have same magnitude and direction• Parallel vecotrs are similarly directed• Antiparallel vectors are oppositely directed

OPERATIONS ON VECTORS

1. Vector Additiona. Parallelogram Method

Note that the vectors are drawn tail-to-tail

• Polygon Method

Note that the vectors are drawn head-to-tail.

• In both vectors, there is a need to specify the drawing scale.• The sum of the vectors is also called the resultant.• The vector equal in magnitude but opposite in direction to the

resultant is called the equilibriant.• Properties of vector addition

a. Closureb. Commutativityc. Associativityd. Identity element is the zero vector, 0 (has zero magntiude,

and no direction)

2. Scalar Multiplication

• The scalar multiple cA is a vector with magnitude c times that of vector of A

• If c is positive, the direction is unchanged. If c is negative, the direction reverses.

• The negative of a vector is a scalar multiple of the vector

Check Questions

1. The magnitudes of forces A and B are 3 N and 4 N respectively. What is the a. Maximum possible magnitude of A + B?b. Minimum possible magnitude of A + B?

2. Is it possible fora. 2 vectors of equal magnitudes to add up to zero?b. 2 vectors of unequal magnitudes to add up to zero?c. 3 vectors of equal magnitudes to add up to zero?d. 3 vectors of unequal magnitudes to add up to zero?

3. Can the sum of the magnitudes of 2 vectors be equal to the magnitude of the sum of the same two vectors? If no, why not? If yes, when?

Subtraction of Vectors

• Addition of a vector with the negative of the other

• Example: Two displacements A and B lie in the xy-plane. A is 50 cm and at 450 from + x-

axis , while B is 30 cm and 1200 from the +x – axis. Find:

a. A + B b. A – B

8/59 Draw vectors D and E such that D = A + B and E = A + C

10/58. What is the vector sum of the 3 forces if each grid square is

2 N on a side?

11/58What is the vector sum A + B + C if each grid is 2 N on

a side?

Example Exercises

2/76Vector A has a magnitude of 8 units and makes an angle of 45 0 with the +x-axis. Vector B also has a magnitude of 8 units and is directed along the –x-axis. Use the graphical methods to find the resultant of the two vectors.

8/76A jogger runs 100 m due west, then changes direction for the second leg of the run. At the end of the run, she is 175 m away from the starting point at an angle of 150 north of west. What were the direction and length of her second displacement?

9/76A man lost in a maze makes three consecutive displacements so that at the end of his travel he is right back where he started. The first displacement if 8 m westward, and the second is 13 m northward. Use the graphical method to find the magnitude and direction of the third displacement.4/76

4/76Each of the displacement vectors A and

B has a magnitude of 3 m. Finda. A + Bb. A – Bc. A – 2B

Check Questions1. The magnitudes of two vectors A and B are 12 units and 8

units, respectively. What are the largest and smallest possible values for the magnitude of the resultant vector R = A + B ?

2. Vectors A and B satisfy the vector equation A + B = 0. a. How does the magnitude of B compare with the magnitude of A? b. How does the direction of B compare with the direction of A?3. Vectors A, B, and C satisfy the equation A + B = C, and their

magnitudes are related by the scalar equation: A2 + B 2 = C 2. How is the vector A oriented with respect to vector B ?

4. Suppose two vectors are added. Under what conditions would the sum of the magnitudes of the vectors equal the magnitude of the resultant vectors?

5. Vector A has a magnitude of 29 units and points in the +y-direction. When vector B is added to A, the resultant vector A + B points in the -y-direction with a magnitude of 14 units. Find the magnitude and direction of vector B.

COMPONENTS OF A VECTOR

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x y

x y

A A A

A A A

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components of v

.

are .

and

ector

uni t vect oe rs ar

x y

x y

A iA jA

A A A

i j

• Component vectors are vectors, while components of a vector are scalars, hence maybe negative, zero or positive.

• For practical reasons, only the x, y and z components of a vector are considered.

• Unless otherwise stated, components are scalars.

,x y A 2 1 2 1,x x y y A

cos

sin

x A

y A

1tan y

x

A

A

2 22 1 2 1A x x y y

1 2 1

2 1

tany y

x x

What are the signs of each of the components of the given vector?

ANALYTICAL METHODS FOR FINDING SUM OF VECTORS

1. Component Method

If:

then

x x xR A B

R =A+B

y y yR A B

2 2

1

tan

x y

y

x

R R R

R

R

2. Cosine Law

2 2 2Cosine law: 2 cosR A B AB

Sine law: sin sin sin

a b c

A B C

Example Exercises42/23. (Cutnell) Two forces are applied to a tree stump to pull it out of the ground.

Force A has a magnitude of 2,240 N and points 34 0 south of east, while force B has a magnitude of 3,160 N and points due south. Using the component method, find the direction and magnitude of the resultant force applied to the stump.

44/23. On a safari, a team of naturalists sets out toward a research station

located 4.8 km away in a direction 42 0 North of East. After traveling in a straight line for 2.4 km, they stop to discover that they have been traveling 22 0 North of East because their guide misread his compass. What are the magnitude and direction (relative to due east) of the displacement vector now required to bring the team to the research station?

48/24. The route followed by a hiker consists of three

displacement vectors A, B and C. Vector A is along a measured trail and is 1,550 m in a direction 25 0 north of east. Vector B is not along a measured trail, but the hiker uses a compass and knows that the direction is 41 0 east of south. Similarly, the direction of vector C is 35 0 north of west. The hiker ends up back where she started, so the resultant displacement A + B + C = 0. Find the magnitudes of

a. vector Bb. vector C

1. A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are: 4 m to the north, 2 m 450 north of east, and 1 m at 30 0 west of south. Starting at the same initial point, an expert golfer could make the hole in what single displacement?

Problem Set

32/23. (Cutnell) Your friend has slipped and fallen. To help her up, you pull with a force F. The vertical component of

this force is 130 N, while the horizontal component is 150 N. Find

a. the magnitude of Fb. the direction

38/23. A force vector points at an angle of 52 0 above the + x-axis. It has a y-component of +290 N. Find:

a. x-component of the force b. the magnitude of the force