Math 152 – Spring 2016
Section 11.2
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Section 11.2 – Vectors and the Dot Product in
Three Dimensions
Definition. Geometrically, a vector is an arrow with both a length and a direction.
Algebraically, a three-dimensional vector is an ordered triple a = ha1 , a2 , a3 i of real
numbers. The numbers a1 , a2 , a3 are call the components of a.
Note.
1. If the origin is the starting point of the vector a = h1, 3, 2i, then the vector a is
the arrow stretching from the origin to the point (1, 3, 2).
2. Vectors do not have to start at the origin. If the vector a = h1, 2, 3i starts at
the point (−2, 3, 1), then the vector a is an arrow stretching from (−2, 3, 1) to
(−2 + 1, 3 + 2, 1 + 3) = (−1, 5, 4).
Definitions.
1. For any point A(x, y, z) and vector a = ha1 , a2 , a3 i, the directed line segment from
A to (x + a1 , y + a2 , z + a3 ) is a representation of the vector a.
2. One of the representations of the vector a = ha1 , a2 , a3 i is the directed line segment
−−→
OP from the origin O to the point P (a1 , a2 , a3 ). The vector a = ha1 , a2 , a3 i is
called the position vector of the point P (a1 , a2 , a3 ).
Theorem. Given the points A(x1 , y1 , z1 ) and B(x2 , y2 , z3 ), the vector a with represen−−→
tation AB is
a = hx2 − x1 , y2 − y1 , z2 − z1 i
Example 1. Find the vector represented by the the directed line segment with initial
point A(−2, 0, 7) and terminal point B(3, −4, 2).
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Section 11.2
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Definition. The magnitude or length of the vector a is the length of any of its
representations and is denoted by the symbol |a| or ||a||. The length of the threedimensional vector a = ha1 , a2 , a3 i is
q
|a| = a21 + a22 + a23
The vector h0, 0, 0i with 0 magnitude is referred to as the zero vector, written 0 or ~0.
Formulas.
1. Vector Addition: If a = ha1 , a2 , a3 i and b = hb1 , b2 , b3 i, then the vector a + b
is defined by
a + b = ha1 , a2 , a3 i + hb1 , b2 , b3 i = ha1 + b1 , a2 + b2 , a3 + b3 i
2. Vector Subtraction: If a = ha1 , a2 , a3 i and b = hb1 , b2 , b3 i, then the vector
a − b is defined by
a − b = ha1 , a2 , a3 i − hb1 , b2 , b3 i = ha1 − b1 , a2 − b2 , a3 − b3 i
3. Scalar Multiplication: If c is a scalar and a = ha1 , a2 , a3 i, then the vector ca
is defined by
ca = cha1 , a2 , a3 i = hca1 , ca2 , ca3 i
Example 2. Let a = h0, −3, 2i and b = h−4, 2, −1i. Find the following:
1. |a| =
2. a + b =
3. a − b =
4. −4b =
5. 2a + 3b =
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Section 11.2
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Definition. A unit vector is a vector whose length is 1. For any vector a = ha1 , a2 , a3 i
1
with a 6= 0, the vector |a|
a is a unit vector with the same direction as a.
Definition. We define three special vectors:
i = h1, 0, 0i
j = h0, 1, 0i
k = h0, 0, 1i
The vectors i, j, and k are unit vectors in the direction of the x-, y-, and z-axes,
respectively.
Theorem. Any three-dimensional vector a can be written as a sum of the vectors i, j,
and k. If a = ha1 , a2 , a3 i, then
ha1 , a2 , a3 i = a1 i + a2 j + a3 k
We call the vectors i, j, and k the standard basis vectors.
Example 3. Find the unit vector in the direction of the vector 3i − 2j + 4k.
Definition. The dot product of two nonzero vectors a and b is the number
a · b = |a||b| cos θ
where θ is the angle between the vectors a and b, 0 ≤ θ ≤ 2π. If either a or b is 0,
then we define a · b = 0.
Example 4. Find a · b if |a| = 6, |b| = 1/3, and the angle between a and b is π/4.
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Section 11.2
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Theorem. The dot product of a = ha1 , a2 , a3 i and b = hb1 , b2 , b3 i is
a · b = a1 b1 + a2 b2 + a3 b3 .
Example 5. Find h5, −3, 2i · h1, −7, −6i
Theorem. If θ is the angle between two nonzero vectors a and b, then
cos θ =
a·b
|a||b|
Example 6. Find the angle between the vectors a = h3, −2, 1i and b = h4, −1, 3i.
Definition. Two vectors are perpendicular or orthogonal if the angle between them
is θ = π/2.
Theorem.
1. Two vectors a and b are orthogonal if and only if a · b = 0.
2. Two vectors a and b are parallel if a = cb for some scalar c.
Example 7. Determine if the following vectors are orthogonal or parallel
1. −5i + 10j − 15k and i − 2j + 3k
2. a = h2, 2, −1i and b = h5, −4, 2i
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Section 11.2
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Definition. The direction angles of a nonzero vector a are the angles α, β, and γ in
the interval [0, π] that makes a with the positive x−, y−, and z-axes.
The direction cosines of the vector a are the cosines of the directions angles, cos α,
cos β, and cos γ.
Theorem. The direction angles are given by the dot product with the standard basis
vectors.
a·i
a·j
a·k
cos α =
cos β =
cos γ =
|a||i|
|a||j|
|a||k|
Example 8. Find the direction angles of the vector a = h1, −3, 0i.
Definition. Let a and b be vectors drawn as in Figure 9, and S be the foot of the
−−→
perpendicular from R to the line containing P Q.
1. The vector projection of b onto a, denoted proja b is the vector with represen−→
tation P S.
2. The scalar projection of b onto a is the number |b| cos θ, where θ is the angle
between a and b (see Figure 10). The scalar projection is also referred to as
the component of b along a and is denoted by compa b. This projection is
negative if π/2 ≤ θ ≤ π.
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Section 11.2
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Theorem.
1. The scalar projection of b onto a is:
compa b =
a·b
|a|
2. The vector projection of b onto a is:
a·b a
a·b
proja b =
a
=
|a|
|a|
|a|2
Example 9. Find the scalar projection and vector projection of b onto a.
1. b = h−1, −2, 2i , a = h3, 3, 4i
2. a = 2i − 3j + k , b = i − 2k
−→
Theorem. If a constant force is a vector F = P R and moves an object from P to Q,
then the work done is
W =F·D
−−→
where D = P Q is the displacement vector.
Example 10. A force is given by a vector F = 2i − j + 3k and moves a particle from
the point P (1, 3, 2) to the point Q(3, 4, 7). Find the work done.