GEOMETRIC VECTORS

Vectors with the same length and same direction are called

equivalent.

If v and w are equivalent, we write:

v = w

GEOMETRIC VECTORS

Two vectors are said to be equal if and only

if they have the same magnitude and

direction.

A

B C

D

ABCD is a parallelogram,

then

and

DCAB

BCAD

GEOMETRIC VECTORS

The vectors of length zero is called the zero vectors, denoted by 0

0 + v = v +0 = v

v+ (-v) = 0

GEOMETRIC VECTORS

(-v) has the same length as v but is

oppositely directed.

Definition:

If v and w are two vectors, then

the difference of w from v is

defined by:

v – w = v + (-w)

GEOMETRIC VECTORS

kv = 0 if k = 0 or v = 0

Vectors in Coordinate Systems

v = (v1, v2)

v1 , v2 = components of v

v = (v1, v2) and w = (w1, w2)

v + w = (v1 + w1, v2 + w2)

Vectors in Coordinate Systems

If v = (v1, v2) and k is a scalar, then :

kv = (kv1, kv2)

Vectors in Coordinate Systems

Vectors in 3-Space: xy –plane, xz-plane, yz-plane

P = (x,y,z) ; x = OX, y = OY, z = OZ

Vectors in Coordinate Systems

Vectors in Coordinate Systems

Vectors in Coordinate Systems

VECTORS OPERATIONS

VECTORS OPERATIONS

VECTORS OPERATIONS

Norms of a Vector

Norms of a Vector

DOT PRODUCT

DOT PRODUCT

Example :

DOT PRODUCT

DOT PRODUCT

DOT PRODUCT

DOT PRODUCT

DOT PRODUCT

DOT PRODUCT

DOT PRODUCT

CROSS PRODUCT

CROSS PRODUCT

Dot Product : Skalar

Cross Product : Vektor

CROSS PRODUCT

Example:

CROSS PRODUCT

Cross Product - Standard Unit Vector

i = (1,0,0), j = (0,1,0), k = (0,0,1)

v = (v1,v2, v3) = v1(1,0,0) + v2 (0,1,0) + v3 (0,0,1)

v = v1i + v2j + v3k

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

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

j x i = -k k x j = -I ix k = -j

Ex : (2,-3,4) = 2i – 3j +4k

Determinant Form of Cross Product

Geometric Interpretation of Cross Product

If u and v are vectors in 3-spaces,

θ : angle between u and v, 0 ≤ θ ≤ π, sin θ ≥ 0,

Geometric Interpretation of Cross Product

Geometric Interpretation of Cross Product

Geometric Interpretation of Cross Product

If u, v, and w are vectors in 3-space, then

is called the scalar triple product of u, v, and w.

u . (v x w)

Geometric Interpretation of Cross Product

Example :

Lines and Places in 3-Spaces

In this section we shall use vectors to derive equations of lines and planes

in 3-space. We shall then use these equations to solve some basic

geometric problems.

Planes in 3-Space

To find the equation of the plane passing through the point P0 (x0,y0,z0)

and having the nonzero vector n = (a,b,c) as a normal.

Vector is orthogonal to n; that is,

We call this the point-normal form of the equation of a plane

Equation of a Plane

Example :

Find an equation of the plane passing through the point (3,-1,7) and perpendicular

to the vector n = (4,2,-5).

Point-normal form is :

4x + 2y – 5z +24 = 0

Theorema :

If a, b, c, and d are constants and a, b, and c are not all zero, then the

graph of the equation ax + by + cz + d = 0

is a plane having the vector n = (a,b,c ) as a normal

ax + by + cz + d = 0 is a linear equation in x, y, and z; it is called the

general form of the equation of a plane

ax + by + cz + d = 0

Equation of the plane is :

Equation of a Plane Through Three Points

Find the equation of the plane passing through the points P1 (1,2,-1), P2 (2,3,1) ,

and P3 (3,-1,2).

ax + by + cz + d = 0