Vectors

VectorsVectors

By Albi Kavo

In Physics we will use two In Physics we will use two different types of different types of

quantities:quantities:•1. Scalar Quantities1. Scalar Quantities- Scalar quantities are quantities with magnitude

but without direction.

•2. Vector Quantities2. Vector Quantities- vector quantities have both direction and

magnitude.

Examples of scalar quantities are:

Examples of vector quantities are:

The direction of a vector is often expressed as an counterclockwise angle of rotation of the vector about its tail from due East (unless specified differently). Using this convention, a vector with a direction of 30 degrees is a vector which has been rotated 30 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector.

Vector quantities are graphically Vector quantities are graphically represented by a represented by a scaled arrow scaled arrow which which has a has a specific directionspecific direction..

AThis is vector A.Magnitude of A = A =|A|

N

EW

S

α

PARALLEL, if:1. They have the same direction.ANTIPARALLEL, if:1. They have opposite direction.EQUAL, if:1.They have the same direction.2.They have the same magnitude.(so two vectors are equal if they are parallel and they have the same magnitude. Their position in space doesn’t matter)

NEGATIVE, if:1.They have opposite direction.2.They have the same magnitude.

AB

A

B

BA

A = B

BA

The addition of vectors is not the same as the addition of scalar quantities.

AB

--- The triangle method ---1.Place the beginning of vector B to the

end of vector A.2.Draw the resultant vector from the

beginning of A to the end of B.

--- The parallelogram method ---1.Place the beginning of vector B to the

beginning of vector A.2.Form a parallelogram

3.Draw the resultant vector from the beginning of the vectors to the intersection of the dotted lines.

A

α

1. Draw the components of the vectors.

2. Find the components of the resultant vector by adding the components together.

Rx=Ax+Bx+CxRy=Ay+By+Cy

B

C

Ax

Cy

By

Bx

Ay

Cx x

y

Finding the length of the components:Ax=A * cos(α)Ay=A * sin(α)

Ax

BxCxRx

Cy

By

Ay

Ry

R

3. Add the components together using the Pythagorean theorem:

R2 = Rx2 + Ry

2

4. Find the direction of R:Θ = tan-1(Ry/Rx)

x

y

z

A unit vector has a magnitude of 1.

îĵ

ќ

x

y

î

ĵ

A

Ax

Ay

We can use unit vectors to simplify writing vector equations.

Ax=Ax î Ay=Ay ĵ

A=Ax î + Ay ĵ

A vector can be multiplied by a number and the result would still be a vector. The resultant vector would have the same direction (n>0) or opposite direction (n<0) to the multiplied vector.

Its magnitude would be n times the magnitude of A: |R|=n * |A|

A

n>0 R has the same direction as An=0 R doesn’t existn<0 R has opposite direction to An=3 |R|=3 * |A|

Example:

R

The resultant of a scalar product of two vectors is a scalar quantity, not a vector.It is also called the dot product of vectors.

A · B=|A|*|B|*cosΘ

The scalar product of:

Perpendicular vectors – is zero because cos90°=0Parallel vectors – is |A|*|B| because cos0°=1

You can use components to calculate the scalar product of vectors.

You can factor as in a normal algebraic expression. Terms like i*i, j*j, k*k, equal 1 (one) because they’re parallel vectors (cos0°=1) and their magnitude is 1 unit.i*j, i*k, j*k equal 0 because they’re perpendicular vectors.

The resultant of a vector product (cross product) of two vectors is a vector. Its magnitude is:

|a x b|=|a|*|b|*sinΘ

It is perpendicular to the plane defined by vectors A and B with

a direction given by the right hand rule.

A x B = -B x A

The magnitude of the vector product of:

Perpendicular vectors – is |a|*|b| because sin90°=1Parallel vectors – is zero because sin0°=0

We can also use components to calculate the magnitude of the vector product.

The vector product of any vector with itself is zero because each vector is parallel to itself and sin(0°)=0.

i x i = j x j = k x k =0Also

i x j = -j x i = k j x k = -k x j = i k x i = -i x k = j

x

y

z

îĵ

ќ

Cx Cy Cz

We can use a determinant to calculate the vector product using components: