Scalar vs Vector

What is the difference between scalar and vector?

Why we have to understand about vector?

Watch this video and tell me what do you think.

Vector Addition

Vector Addition (2)

U

V

U

V

U

V

U+V

U

V

U

V U+V

Parallelogram Rule U + V = V + U

Vector addition is commutative

(U + V) + W = U + (V + W)

Vector addition is associative

When two or more vectors are added, the

order in which they are added doesn’t matter.

U

V

W

U+V+W U

V

W

The sum of

those three

vectors is

zero

Product of a Scalar and a Vector

The product of a scalar a and a vector U is defined by a vector aU with magnitude |a||U|.

Its direction is the same with U when a is positive and when a is negative, its direction is opposite.

The division of U by a is defined by 1

𝑎 U.

U 2U -U 𝟏

𝟐 U

Vector Subtraction The difference of two vectors U and V is defined by:

U – V = U + (-1) V

(-1) V

U

U - V

Unit Vectors Unit vector is a vector whose magnitude is 1.

Any vector U can be expressed as |U|e where e is a unit vector

with the same direction as U.

Dividing a vector U by its magnitude yields a unit vector with

the same direction as U.

U

|U| e

1 𝐔

|𝐔| = e

Components in Two Dimensions

U y

x Ux

Uy

U y

x

Uxi

Uyj

j

i

A vector U that is parallel to

the x-y plane can be expressed

as

U = Uxi + Uyj

The magnitude of U is

|U| = 𝑈𝑥2 + 𝑈𝑦2

Manipulating Vectors in Terms of Components

U + V = (Uxi + Uyj) + (Vxi + Vyj)

= (Ux + Vx) i + (Uy + Vy) j

aU = a(Uxi + Uyj)

= aUxi + aUyj

y

x

U+V

U

V

y

x

(Ux+Vx)i

(Uy+Vy)j U+V U+V

Uxi Uyj

Vyj Vxi

Position Vectors in Terms of Components

y

x

A

B

rAB

(xA, yA)

(xB, yB)

The position vector from A to B is given by

rAB = ( xB – xB)i + (yB – yA)j

Dot Products Dot products are used to :

1. Evaluate moments of forces about points and lines.

2. Determine the components of a vector parallel and

perpendicular to a given line.

3. Determine the angle between two lines in space.

Etc..

Dot Products The dot product of two vectors U and V is defined by

U.V = |U||V| cos θ

where θ is the angle between the vectors when they are placed

tail to tail.

If |U| ≠ 0 and |V| ≠ 0, U.V = 0 if and only if U and V are

perpendicular.

θ

U

V

Dot Products in Terms of Components The dot product of U and V is given in terms of the

components of the vectors by

U.V = UxVx + UyVy

Vector Components Parallel and Normal to a Line

A vector U can be resolved into

1. A vector component Up that is parallel to a given line L

2. A vector component Un that is normal to L.

L

U

θ

Up Un

If e is a unit vector that is parallel to L, the parallel component of U

is given by

Up = (e.U) e The normal component can be obtained from the relation

Un = U - Up

Cross Products Cross product of two vector has many applications, such as:

1. Determining moments of forces.

2. Determining the rate of rotation of a fluid particle.

3. Calculating the force exerted on a charged particle by a

magnetic field.

Cross Products

The cross product of two vectors U and V is defined by

U x V = |U||V| sin ϴ e

The unit vector e is defined to be perpendicular to U and

perpendicular to V and directed so that U, V, e form a right-handed system.

If |U| ≠ 0 and |V| ≠ 0, U x V = 0 if and only U and V are parallel.

Let’s do the exercises on your sheets!