Section 9.4: Vectors and Analytic Geometry
Definition: A vector is a quantity (such as velocity or force) that has both magnitude and
direction. In particular, an n-dimensional vector is an ordered n-tuple of real numbers
~x = hx1 , x2 , . . . , xn i.
The numbers x1 , x2 , . . . , xn are called the components of ~x.
Definition: A representation of a vector is an arrow with an initial and terminal point.
The length of the arrow represents the magnitude of the vector and the arrow points in the
direction of the vector. The representation of a vector whose initial point is at the origin is
called the vector representation or position vector.
Theorem: The vector with initial point A = (a1 , a2 , . . . , an ) and terminal point B = (b1 , b2 , . . . , bn )
is given by
−→
AB = hb1 − a1 , b2 − a2 , . . . , bn − an i.
−→
Example: Find the vector AB with initial point A = (4, −1) and terminal point B = (1, 2).
−→
Draw AB and the corresponding position vector.
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Theorem: (Vector Algebra)
Suppose that ~x = hx1 , x2 , . . . , xn i and ~y = hy1 , y2 , . . . , yn i are vectors and c is a scalar.
1. Scalar Multiplication: The vector c~x is defined by
c~x = hcx1 , cx2 i.
Geometrically, scalar multiplication is represented by scaling the magnitude of ~x and/or
reversing its direction.
2. Vector Addition: The vector ~x + ~y is defined by
~x + ~y = hx1 + y1 , x2 + y2 , . . . , xn + yn i.
Geometrically, vector addition is represented by the Triangle or Parallelogram Law.
3. Vector Subtraction: The vector ~x − ~y is defined by
~x − ~y = hx1 − y1 , x2 − y2 , . . . , xn − yn i.
A geometric representation of vector subtraction is given below.
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Example: Let ~x = h−4, 3, 1i and ~y = h0, −2, 3i.
(a) Find 2~x + 3~y .
(b) Find ~x − ~y .
Definition: The magnitude or length of a vector ~x = hx1 , x2 , . . . , xn i is given by
q
||~x|| = x21 + x22 + · · · + x2n .
The only vector with magnitude 0 is called the zero vector and is denoted by ~0.
Example: Find the magnitude of ~x = h−2, 1, 3i.
Definition: A unit vector is a vector with magnitude one. If ~x 6= ~0, then a unit vector in
the direction of ~x is
~x
1
x̂ =
=
~x.
||~x||
||~x||
The process of finding a unit vector in the direction of a given vector is called normalization.
Example: Find a unit vector in the direction of ~x = h6, 3, −6i.
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Suppose that the vector ~v = hx, yi has magnitude r and forms an angle θ measured
counterclockwise from the positive x-axis.
It follows that
cos θ =
x
r
sin θ =
x = r cos θ
y
r
y = r sin θ.
Therefore, the vector ~v is given by
~v = hr cos θ, r sin θi.
Example: Find the components of the vector ~x that has magnitude 4 and forms an angle of
60◦ measured counterclockwise from the positive x-axis.
Example: Find the components of the vector ~x that has magnitude 2 and forms an angle of
150◦ measured clockwise from the positive x-axis.
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Definition: The dot product of two vectors ~x = hx1 , x2 , . . . , xn i and ~y = hy1 , y2 , . . . , yn i is
the scalar
n
X
~x · ~y =
xk yk = x1 y1 + x2 y2 + · · · + xn yn .
k=1
Example: Find the dot product of ~x = h2, 3, −1i and ~y = h−3, 5, 4i.
Note: The dot product can be used to find the magnitude of a vector ~x = hx1 , x2 , . . . , xn i.
In particular,
√
||~x|| = ~x · ~x.
Example: Use the dot product to find the magnitude of ~x = h−1, 4, 3i.
Theorem: (Angle Between Two Vectors)
If θ is the angle between two nonzero vectors ~x and ~y , then
cos θ =
~x · ~y
.
||~x||||~y ||
Example: Find the angle between the vectors ~x = h1, 2, 2i and ~y = h3, 4, 0i.
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Example: The points A = (3, 0), B = (5, 6), and C = (−2, 1) form a triangle. Find the angle
located at the vertex A.
Definition: Two vectors ~x and ~y are parallel if there exists a scalar c such that ~x = c~y . Two
vectors are orthogonal or perpendicular if the angle between them is θ = π/2 radians.
Theorem: (Orthogonal Vector Theorem)
Two nonzero vectors ~x and ~y are orthogonal if and only if
~x · ~y = 0.
Example: Determine whether the given vectors are orthogonal, parallel, or neither.
(a) ~x = h1, 5, −2i and ~y = h3, 1, 4i
(b) ~x = h2, −1, 3i and ~y = h−4, 2, −6i
(c) ~x = h1, 0, −2i and ~y = h2, −1, 3i
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Example: Find all values of x such that ~v = hx, 2xi and w
~ = hx, −2i are orthogonal.
A line in the plane can be defined by a point and a vector. Suppose that we want to find
~ = hA, Bi.
an equation of the line passing through (x0 , y0 ) and perpendicular to N
If (x, y) is another point on the line, then the vector ~v = hx − x0 , y − y0 i lies on the line.
Therefore,
~ · ~v = 0
N
A(x − x0 ) + B(y − y0 ) = 0.
Theorem: (Equation of a Line)
An equation of the line passing through (x0 , y0 ) and perpendicular to hA, Bi is
A(x − x0 ) + B(y − y0 ) = 0.
Example: Find an equation of the line passing through (3, 2) and perpendicular to h−1, 1i.
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A plane in space can be defined by a point and a vector. Suppose that we want to find
~ = hA, B, Ci.
an equation of the plane passing through (x0 , y0 , z0 ) and perpendicular to N
𝑧
𝑁
𝑣
𝑦
𝑥
If (x, y, z) is another point on the plane, then ~v = hx − x0 , y − y0 , z − z0 i lies in the plane.
Therefore,
~ · ~v = 0
N
A(x − x0 ) + B(y − y0 ) + C(y − y0 ) = 0.
Theorem: (Equation of a Plane)
An equation of the plane containing the point (x0 , y0 , z0 ) and perpendicular to hA, B, Ci is
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0.
Example: Find an equation of the plane passing through (3, −1, 2) and perpendicular to
h−1, 1, 2i.
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A line in the plane or in space can be defined by a point and a vector. Suppose that we
want to find an equation of the line passing through (x0 , y0 , z0 ) and parallel to the vector
~v = hA, B, Ci.
𝑧
𝑣 = 𝐴, 𝐵, 𝐶
𝑥0 , 𝑦0 , 𝑧0
𝑦
𝑅 𝑡 = 𝑥0 + 𝐴𝑡, 𝑦0 + 𝐵𝑡, 𝑧0 + 𝐶𝑡
𝑥
Definition: The line containing the point (x0 , y0 , z0 ) and parallel to the vector ~v = hA, B, Ci
has parametric equations
x = x0 + At,
y = y0 + Bt,
z = z0 + Ct,
where t ∈ R is a parameter. These equations can be expressed in vector form as
~
R(t)
= hx0 + At, y0 + Bt, z0 + Cti.
Example: Find parametric equations of the line in the plane passing through A = (2, 1) and
B = (3, 5). Then eliminate the parameter to find a Cartesian equation of the line.
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Example: Find parametric equations of the line in space passing through the points A =
(1, 0, 4) and B = (3, 2, 0).
~ = h−1, 1, 3i
Example: Consider the plane passing through (2, 0, −1) and perpendicular to N
as well as the line passing through A = (1, 0, −2) and B = (−1, −1, 1). Determine where
the plane and line intersect.
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