VECTORS

VECTORSIntroduction to vectorsIn physics and other sciences 2 different quantities can be found: - scalars and vectors.

A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it.Magnitude A numerical value with units.

Scalar ExampleMagnitudeSpeed20 m/sDistance10 mAge15 yearsA VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. We usuaLly represent a vector with an arrow:

The direction of the arrow is the direction of the vector, the length is the magnitude. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

VectorMagnitude & DirectionVelocity20 m/s, NAcceleration10 m/s/s, EForce5 N, West

Operations with vectors( geometric intepretation )

1. Multiples of Vectors - Given a real number c, we can multiply a vector by c by multiplying its magnitude by c:v2v-2vNotice that multiplying a vector by a negative real number reverses the direction.2. ADDITION: When two vectors point in the SAME direction, simply add them together.EXAMPLE: A man walks 46.5 m east, then another 20 m east. Calculate his displacement relative to where he started. 66.5 m, E46.5 m, E+20 m, E Two vectors can be added using the Parallelogram Lawuvu + v3. SUBTRACTION: When two vectors point in the OPPOSITE direction, simply subtract them.EXAMPLE: A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started. 26.5 m, E46.5 m, E-20 m, Wuv We can subtract 2 vectors using the paralelogram rule:u - vVectors in plane

To do computations with vectors, we place them in the plane and find their components.v(2,2)(5,6)The initial point is the tail, the head is the terminal point. The components are obtained by subtracting coordinates of the initial point from those of the terminal point. The first component of v is 5 -2 = 3. The second is 6 -2 = 4. We write v = The magnitude of the vector is the length of the segment, it is written ||v||.|v| = ( v12 + v22 ) Once we have a vector in component form, the arithmetic operations are easy.

1. To multiply a vector by a real number, simply multiply each component by that number.

Example: If v = , -2v =

2. To add vectors, simply add their components.

For example, if v = and w = , then v + w = .

We considered vectors in THE RECTANGULAR (CARTESIAN) PLANE. They have been 2 dimensional, but vectors work perfectly well in 3 or more dimensions:

Examples:

1. Add the vectors a = (3,7,4) and b = (2,9,11)c = a + bc = (3,7,4) + (2,9,11) = (3+2,7+9,4+11) = (5,16,15)

2. What is the magnitude of the vector w = (1,-2,3) ? |w| = ( 12 + (-2)2 + 32 ) = ( 1+4+9 ) = 14

Multiplying a Vector by a Vector (Dot Product and Cross Product) How do you multiply two vectors together? There is more than one way!

1. The scalar or Dot Product (the result is a scalar). 2. The vector or Cross Product (the result is a vector).

1. Dot product of vectors a and b:

You can calculate the Dot Product of two vectors this way:a b = |a| |b| cos()

OR you can calculate it this way:In 2 dimensions:

In 3 dimensions: a b = ax bx + ay by + az bz

The Dot Product gives a scalar (ordinary number) answer, and is sometimes called the "scalar product"

a b = ax bx + ay by