geometric vectors - i'm dafiqurrohman · geometric vectors (-v) has the same length as v but...
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GEOMETRIC VECTORS
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Vectors with the same length and same direction are called
equivalent.
If v and w are equivalent, we write:
v = w
GEOMETRIC VECTORS
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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
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GEOMETRIC VECTORS
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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)
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GEOMETRIC VECTORS
kv = 0 if k = 0 or v = 0
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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)
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Vectors in Coordinate Systems
If v = (v1, v2) and k is a scalar, then :
kv = (kv1, kv2)
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Vectors in Coordinate Systems
Vectors in 3-Space: xy –plane, xz-plane, yz-plane
P = (x,y,z) ; x = OX, y = OY, z = OZ
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Vectors in Coordinate Systems
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Vectors in Coordinate Systems
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Vectors in Coordinate Systems
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VECTORS OPERATIONS
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VECTORS OPERATIONS
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VECTORS OPERATIONS
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Norms of a Vector
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Norms of a Vector
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DOT PRODUCT
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DOT PRODUCT
Example :
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DOT PRODUCT
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DOT PRODUCT
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DOT PRODUCT
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DOT PRODUCT
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DOT PRODUCT
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DOT PRODUCT
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DOT PRODUCT
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CROSS PRODUCT
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CROSS PRODUCT
Dot Product : Skalar
Cross Product : Vektor
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CROSS PRODUCT
Example:
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CROSS PRODUCT
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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
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Determinant Form of Cross Product
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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,
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Geometric Interpretation of Cross Product
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Geometric Interpretation of Cross Product
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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)
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Geometric Interpretation of Cross Product
Example :
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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
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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 :
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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