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Section 11.2: Vectors and the Dot Product in Three Dimensions
Definition: A three dimensional vector is an ordered triple a = ha1 , a2 , a3 i of real numbers.
−
−
→
Definition: Given the points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ), the vector a with representation AB is
a = hx2 − x1 , y2 − y1 , z2 − z1 i
−
−→
−
−→
Example: For the points, A(1, 2, 8) and B(4, 7, 2), find AB and BA.
Definition: Let a = ha1 , a2 , a3 i, b = hb1 , b2 , b3 i and c be a scalar, i.e. c ∈ ℜ.
Scalar Multiplication: ca = c ha1 , a2 , a3 i = hca1 , ca2 , ca3 i
Length or magnitude of a is |a| =
q
a21 + a22 + a23
Vector Addition: a + b = ha1 + b1 , a2 + b2 , a3 + b3 i
Zero vector: 0 = h0, 0, 0i and |0| = 0
Definition: Two vectors are parallel if one vector is a scalar multiple of the other. i.e. there exists a
c ∈ ℜ such that ca = b.
Definition: A vector of length 1 is called a unit vector.
To find a unit vector in the same direction as a, divide vector a by its magnitude. This process is
called normalizing a.
Example: Find the following using these vectors: a = h1, 2, 4i, b = h3, 1, −2i, and c = h2, −4, 1i.
A) 2a + 3b =
B) a − 2c =
C) Find a vector of length 3 in the opposite direction of a.
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Definition: The dot product of two nonzero vectors a and b is the number
a &middot; b = |a||b| cos θ
where θ is the angle between the vectors a and b, 0 ≤ θ ≤ π. If either a or b is 0, then a &middot; b = 0.
The dot product of a = ha1 , a2 , a3 i and b = hb1 , b2 , b3 i is
a &middot; b = a1 b1 + a2 b2 + a3 b3
Definition: Two non-zero vectors a and b are orthogonal if and only if a &middot; b = 0, i.e. the angle
between them is π/2.
Example: Find the following using these vectors: a = h−1, −2, −3i, b = h−10, 2, 1i, and c = h2, 8, −6i.
A) a &middot; b =
B) a &middot; c =
C) Find the angle between a and b.
Example: If |a| = 1 and |b| = 2, what is the maximum for a &middot; b? What does this say about the
vectors?
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Directional angles/and Direction Cosines
Example: Find the direction angles for a = h1, 0, 5i
Projections
Scalar projection of b onto a:
compa b =
a&middot;b
|a|
Vector projection of b onto a:
proja b =
a&middot;b
a
|a|2
Example: Find the vector and scalar projections of m = h2, 1, 5i onto n = h1, 2, 3i
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