Elementary Linear Algebra Anton

1
Elementary Linear AlgebraAnton Rorres, 9th
Edition

• Lecture Set 03
• Chapter 3
• Vectors in 2-Space and 3-Space

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Chapter Content

• Introduction to Vectors (Geometric)
• Norm of a Vector Vector Arithmetic
• Dot Product Projections
• Cross Product
• Lines and Planes in 3-Space

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Definitions

• If v and w are any two vectors, then the sum v
  w is the vector determined as follows
• Position the vector w so that its initial point
  coincides with the terminal point of v. The
  vector v w is represented by the arrow from the
  initial point of v to the terminal point of w.
• If v and w are any two vectors, then the
  difference of w from v is defined by v w v
  (-w).
• If v is a nonzero vector and k is nonzero real
  number (scalar), then the product kv is defined
  to be the vector whose length is k times the
  length of v and whose direction is the same as
  that of v if k gt 0 and opposite to that of v if k
  lt 0. We define kv 0 if k 0 or v 0.
• A vector of the form kv is called a scalar
  multiple.

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Remarks

• Rectangular coordinate systems in 3-space fall
  into two categories, left-handed and right-handed.

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Translation of Axes

• In the figure we have translated the axes of an
  xy-coordinate system to obtain an x?y?-coordinate
  system whose O? is at point (x, y) (k,l).
• A point P in 2-space now has both (x, y)
  coordinates and (x?, y?) coordinates.
• x? x k, y? y l, these formulas are called
  the translation equations.
• In 3-space the translation equations are
• x? x k, y? y l, z? z m,
• where (k, l, m) are the xyz-coordinates of the
  x?y?z?-origin.

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Theorem 3.2.1 (Properties of Vector Arithmetic)

• If u, v and w are vectors in 2- or 3-space and k
  and l are scalars, then the following
  relationships hold.
• u v v u
• (u v) w u (v w)
• u 0 0 u u
• u (-u) 0
• k(lu) (kl)u
• k(u v) ku kv
• (k l)u ku lu
• 1u u

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Norm of a Vector

• The length of a vector u is often called the norm
  of u and is denoted by u.
• It follows from the Theorem of Pythagoras that
  the norm of a vector u (u1,u2,u3) in 3-space is
  
• A vector of norm 1 is called a unit vector.
• The distance between two points is the norm of
  the vector.
• The length of the vector ku ku k u.

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Definitions

• Let u and v be two nonzero vectors in 2-space or
  3-space, and assume these vectors have been
  positioned so their initial points coincided. By
  the angle between u and v, we shall mean the
  angle ? determined by u and v that satisfies 0 ?
  ? ? ?.
• If u and v are vectors in 2-space or 3-space and
  ? is the angle between u and v, then the dot
  product or Euclidean inner product u v is
  defined by

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Example

• If the angle between the vectors u (0,0,1) and
  v (0,2,2) is 45?, then

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Theorems

• Theorem 3.3.1
• Let u and v be vectors in 2- or 3-space.
• v v v2 that is, v2 (v v)½
• If the vectors u and v are nonzero and ? is the
  angle between them, then
• ? is acute if and only if u v gt 0
• ? is obtuse if and only if u v lt 0
• ? ?/2 if and only if u v 0
• Theorem 3.3.2 (Properties of the Dot Product)
• If u, v and w are vectors in 2- or 3-space, and k
  is a scalar, then
• u v v u
• u (v w) u v u w
• k(u v) (ku) v u (kv)
• v v gt 0 if v ? 0, and v v 0 if v 0

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Orthogonal Vectors

• Definition
• Perpendicular vectors are also called orthogonal
  vectors.
• Two nonzero vectors are orthogonal if and only if
  their dot product is zero.
• To indicate that u and v are orthogonal vectors
  we write
• u?v.
• Example
• Show that in 2-space the nonzero vector n (a,b)
  is perpendicular to the line ax by c 0.

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An Orthogonal Projection

• To "decompose" a vector u into a sum of two
  terms, one parallel to a specified nonzero vector
  a and the other perpendicular to a.
• We have w2 u w1 and w1 w2 w1 (u w1)
  u
• The vector w1 is called the orthogonal projection
  of u on a or sometimes the vector component of u
  along a, and denoted by projau
• The vector w2 is called the vector component of u
  orthogonal to a, and denoted by w2 u projau

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Theorem 3.3.3

• If u and a are vectors in 2-space or 3-space and
  if a ? 0, then

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Example

• Solution

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Example

• Solution

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Example

• Solution (cont.)

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Cross Product

• Definition
• If u (u1, u2, u3) and v (v1, v2, v3) are
  vectors in 3-space, then the cross product u ? v
  is the vector defined byor in determinant
  notation
• Example
• Find u ? v, where u (1, 2, -2) and v (3, 0,
  1).

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Theorems

• Theorem 3.4.1 (Relationships Involving Cross
  Product and Dot Product)
• If u, v and w are vectors in 3-space, then
• u (u ? v) 0
• v (u ? v) 0
• u ? v u2v2 (u
  v)2 (Lagranges identity)
• u ? (v ? w) (u w) v (u v)
  w (relationship between cross dot product)
• (u ? v) ? w (u w) v (v w) u (relationship
  between cross dot product)
• Theorem 3.4.2 (Properties of Cross Product)
• If u, v and w are any vectors in 3-space and k is
  any scalar, then
• u ? v - (v ? u)
• u ? (v w) u ? v u ? w
• (u v) ? w u ? v v ? w
• k(u ? v) (ku) ? v u ? (kv)
• u ? 0 0 ? u 0
• u ? u 0

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Standard Unit Vectors

• The vectors
• i (1,0,0), j (0,1,0), k (0,0,1)
• have length 1 and lie along the coordinate axes.
  They are called the standard unit vectors in
  3-space.
• Every vector v (v1, v2, v3) in 3-space is
  expressible in terms of i, j, k since we can
  write
• v (v1, v2, v3) v1(1,0,0) v2 (0,1,0) v3
  (0,0,1) v1i v2j v3k
• For example, (2, -3, 4) 2i 3j 4k
• Note that
• i ? i 0, j ? j 0, k ? k 0
• i ? j k, j ? k i, k ? i j
• j ? i -k, k ? j -i, i ? k -j

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Cross Product

• A cross product can be represented symbolically
  in the form of 3?3 determinant
• Geometric interpretation of cross product
• From Lagranges identity, we have

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Area of a Parallelogram

• Theorem 3.4.3 (Area of a Parallelogram)
• If u and v are vectors in 3-space, then u ? v
  is equal to the area of the parallelogram
  determined by u and v.
• Example
• Find the area of the triangle determined by the
  point (2,2,0), (-1,0,2), and (0,4,3).

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Triple Product

• Definition
• If u, v and w are vectors in 3-space, then u (v
  ? w) is called the scalar triple product of u, v
  and w.
• Remarks
• The symbol (u v) ? w make no sense.
• u (v ? w) w (u ? v) v (w ? u)

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Theorem 3.4.4

• The absolute value of the determinantis equal
  to the area of the parallelogram in 2-space
  determined by the vectors u (u1, u2), and v
  (v1, v2),
• The absolute value of the determinantis equal
  to the volume of the parallelepiped in 3-space
  determined by the vectors u (u1, u2, u3), v
  (v1, v2, v3), and w (w1, w2, w3),

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Remark
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Theorem 3.4.5

• If the vectors u (u1, u2, u3), v (v1, v2,
  v3), and w (w1, w2, w3) have the same initial
  point, then they lie in the same plane if and
  only if

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Planes in 3-Space

• One can specify a plane in 3-space by giving its
  inclination and specifying one of its points.
• A convenient method for a plane is to specify a
  nonzero vector, called a normal, that is
  perpendicular to the plane.
• The point-normal form of the equation of a plane
• n (a, b, c)
• a(x-x0) b(y-y0) c(z-z0) 0

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Example
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Theorem 3.5.1

• 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.
• Remark
• The above equation is a linear equation in x, y,
  and z it is called the general form of the
  equation of a plane.
• Theorem 3.5.2 (Distance between a Point and a
  Plane)
• The distance D between a point P0(x0,y0,z0) and
  the plane ax by cz d 0 is

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The Solution of a System in 3-Space
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Example
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Example
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Line in 3-Space

• Suppose that l is the line in 3-space through the
  point P0(x0,y0,z0) and parallel to the nonzero
  vector v (a, b, c).
• l consists precisely of those points P0(x0,y0,z0)
  for which the vector is parallel to v,
  that is, for which there is a scalar t such that
• Parametric equations for l

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Example

• Parametric equations of a line

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Example (Intersection of a Line and the xy-Plane)
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Example (Line of Intersection of Two Planes)
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Example

• A line parallel to a given vector

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Theorem 3.5.2 (Distance Between a Point and a
Plane)

• The distance D between a point P0(x0,y0,z0) and
  the plane ax by cz d 0 is

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Example (Distance Between a Pont and a Plane)
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Example (Distance Between Parallel Planes)