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Vectors

More math concepts

Objectives

• Distinguish between vector and scalar quantities.

• Carry out addition and scalar multiplication of vectors.

• Understand forces as vectors.

What’s the Point?

• How can we specify quantities that depend on direction?

• How do forces combine?

Vectors and Scalars

• Vector: quantity needing a direction to fully specify (direction + magnitude)

• Scalar: directionless quantity

Arrows for Vectors

direction: obvious

magnitude: length

location is irrelevant

these are identical

Represent as Components

Components: projections in (x, y) directions

BA

A = (4, 3)

B = (0, –2)

xy

Magnitude from Components

Components: lengths of sides of right triangle

Magnitude: length of hypotenuse

A

A = (4, 3)

||A ||= A = 42 + 32

Physics Vectors and Scalars

• Position, displacement, velocity, acceleration, and force are vector quantities.

• Mass and time are scalar quantities.

• (Yes, there are many others)

Combine Displacement Vectors

(CR to HA) + (HA to Union) = (CR to Union)

Add Vectors

A

C B

A + B = C

Head-to-tail (not in your book)

A

B

How to Add Vectors

• Place following vector’s tail at preceding vector’s head

• Resultant starts where the first vector starts and ends where the last vector ends

• Add any number of vectors, one after another

Sum by Components

Vector sum: Add (x, y) components individually

C

BA

A = (4, 3)

B = (0, –2)

C = A + B = (4+0, 3–2) = (4, 1)

Poll Question

Which vector is the sum of vectors A and B?

A

DC

B

AB

Group Work

1. Draw two vectors A and B. Graphically find:

• A + B

Poll Question

Is vector addition commutative?

A. Yes.

B. No.

Vector Addition is Commutative

A + B = CA

BB + A = C

A + B = B + A

Add Vectors

Book uses parallelogram rule

emphasizes commutativity

Respect the Units

• For a vector sum to be meaningful, the vectors you add must have the same units!

5 s + 10 s = 15 s

5 kg + 10 m = 15 ?

• Or, algebra in general:5 a + 10 a = 15 a

5 b + 10 c = 15 ?

good!

Bad!

good!

Bad!

• Just as with scalars:

Subtract Vectors

A

B

Add the negative of the vector being subtracted.

–B

A – B = A + (–B) = D

D

–BA

(Negative = same magnitude, opposite direction: what you must add to get zero)

Group Work

2. Make up three vectors A, B, and C. Graphically show:

• A – B• A + B + C• C + A + B

Multiplication by a Scalar

• Product of (scalar)(vector) is a vector

• The scalar multiplies the magnitude of the vector; direction does not change

• Direction reverses if scalar is negative

A 2 A 1/2 A–2 A

Scalar Multiplication Example

Velocity (a vector) time (a scalar)

v t = r

Result is displacement (a vector).

The vectors are in the same direction, but have different units!

Net Force

• Forces on an object add together.

• Forces can oppose each other.• Net force is the vector sum of all forces

acting on a body.• The net force on a body at rest is zero.

Poll: Hammock Example

A hammock slung between trees 8 m apart sags 1 m when a person lies in it.

The net force acting on the person is

A. Equal to the weight of the person.

B. Equal to the tension in one cable.

C. Zero.

D. There is not enough information to answer.

8 m

1 m

weight

FF

Working with Commonly-Encountered Forces

Tension

Tension Forces

• In cables, threads, chains, etc.

• Direction: along the cable, inward

Poll: Hammock Tension

A hammock slung between trees 8 m apart sags 1 m when a person lies in it.

The tension in a cable is

A. Equal to the weight of the person.

B. About half the weight of the person.

C. Zero.

D. Much more than the weight of the person.

E. There is not enough information to answer.

8 m

1 m

weight

FF

Hammock Forces

weight

tensiontension

forces add to zero

Tension exceeds weight for a shallow angle!

Application: Lumbar Forces

Spinal curvature

standing sitting

Application: Lumbar Forces

Reaching with a load

standing sitting

weight weight

Application: Lumbar Forces

Standing

torque

supporttens

ion

Application: Lumbar Forces

Sitting

torque

supporttens

ion

huge!

Reading for Next Time

• Force, mass, and acceleration: how and why motion changes

• Keep in mind how this applies in everyday experience.