Dot Product

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Section 3.3

• Dot Product
• Projections

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THE DOT PRODUCT
If u and v are vectors in 2- 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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COMPONENT FORM OF THE DOT PRODUCT
If u (u1, u2) and v (v1, v2) are vectors in
2-space, then u v u1v1 u2v2 If u (u1,
u2, u3) and v  (v1, v2, v3) are vectors in
3-space, then u v u1v1 u2v2 u3v3
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ANGLE BETWEEN VECTORS
If u and v are nonzero vectors and ? is the angle
between them, then the angle between them is
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DOT PRODUCT, NORM, AND ANGLES
Theorem 3.3.1 Let u and v be vectors in 2- or
3-space. (a) v v v2 that is, v (v
v)1/2 (b) 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 ? p/2 if and only if u
v 0
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ORTHOGONAL VECTORS
Perpendicular vectors are also called orthogonal
vectors. Two nonzero vectors are orthogonal if
and only if their dot product is zero. We
consider the zero vector orthogonal to all
vectors. To indicate that u and v are orthogonal,
we write
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PROPERTIES OF THE DOT PRODUCT
If u, v, and w are vectors is 2- or 3-space and k
is a scalar, then (a) u v v u (b) u (v
w) u v u w (c) k(u v) (ku) v u
(kv) (d) v v gt 0 if v ? 0, and v v 0 if v
0
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ORTHOGONAL PROJECTION
Let u and a ? 0 be positioned so that their
initial points coincide at a point Q. Drop a
perpendicular from the tip of u to the line
through a, and construct the vector w1 from Q to
the foot of this perpendicular. Next form the
difference w2 u - w1. The vector w1 is called
the orthogonal projection of u on a or sometimes
the vector component of u along a. It is denoted
by proja u. The vector w2 is called the vector
component of u orthogonal to a and can be written
as w2 u - proja u.
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THEOREM
Theorem 3.3.3 If u and a are vectors in 2-space
or 3-space and if a ? 0, then
(vector component of u along a)
(vector component of u orthogonal to a)
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NORM OF THE ORTHOGONAL PROJECTION OF u ALONG a
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DISTANCE BETWEEN A POINT AND A LINE
The distance D between a point P(x0, y0) and the
line ax by c 0 is given by the formula