Jim Lambers
MAT 169
Fall Semester 2009-10
Lecture 18 Notes
These notes correspond to Section 10.2 in the text.
Vectors
When an object is displaced, it is moved a certain distance, and also in some direction. Both the
distance and direction are required in order to fully describe the object’s motion. Similarly, an
object’s velocity incorporates the rate at which it is traveling, and its direction of travel. A vector
refers to a quantity, such as displacement or velocity, that has, as properties, both magnitude and
direction.
Visually, a vector is represented by an arrow. The length of the arrow indicates the magnitude
of the vector, and the direction of the arrow is the direction of the vector. The point at the tail
of the arrow is called the initial point of the vector, and the tip of the arrow is called the terminal
point. A typical vector is shown in Figure 1.
Two arrows that are pointing in the same direction, and the same length, but are located at
different positions, represent vectors that are indistinguishable from one another, because they have
the same magnitude and direction. We say that these vectors are equivalent, or equal.
We will normally denote vectors by bold lowercase letters, such as u or v. A vector that is of
particular importance is the zero vector, denoted by 0. It is a vector whose magnitude, or length,
is zero. As such, it does not have any particular direction.
Combining Vectors
Let u and v be two vectors that are positioned in such a way that the initial point of v coincides
with the terminal point of u. The sum of u and v is the vector u + v whose initial point is the
initial point of u, and whose terminal point is the terminal point of v.
The vectors u, v and u+v form a triangle, which is why this definition of the sum of two vectors
is called the Triangle Law. This law is illustrated in Figure 2(a). However, if the two vectors are
added in reverse order, the sum v + u turns out to be equal to u + v. That is, addition of vectors is
commutative. To see this, we note that if the initial point of u in the sum u + v coincides with the
initial point of v in the sum v + u, the two copies of u and v form a parallelogram, with the two
sums coinciding with one of its diagonals. Therefore, commutativity of vector addition is called the
Parallelogram Law, which is illustrated in Figure 2(b).
Another important operation that can be performed on vectors is called scalar multiplication.
This operation entails scaling the magnitude of a vector u by a number 𝑠, which, in this context,
is called a scalar. The direction of the resulting vector, denoted by 𝑠u and called a scalar multiple
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Figure 1: A representation of a vector by an arrow, which indicates the length and direction of the
vector.
of u, is the same as that of u if 𝑠 > 0, and the opposite of that of u if 𝑠 < 0. If 𝑠 = 0, 𝑠u = 0, the
zero vector.
Two vectors that have the same direction are said to be parallel. It follows from the definition
of scalar multiplication that two vectors are parallel if and only if one is a scalar multiple of the
other. One scalar multiple of particular importance is the negative of a vector. Given a vector u,
its negative is the vector −u obtained by scaling u by −1. Its direction is the opposite of that of
u, but its magnitude is the same.
Example Figure 3 displays a vector u and some examples of scalar multiples of u, including 2u,
−u, and −3u. □
The negative is used to define the operation of vector subtraction. Given two vectors u and v,
we define u − v to be the sum u + (−v). This operation gives us another way of characterizing
equivalent vectors: two vectors are equivalent, or equal, if and only if their difference is the zero
vector: u − v = 0 implies u = v.
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Figure 2: (a) The Triangle Law. (b) The Parallelogram Law.
Components
So far, we have defined vectors, and operations on them, geometrically, in terms of vectors’ magnitude and direction. However, in many situations, it is easier to treat vectors algebraically. This
requires the introduction of a coordinate system. In the three-dimensional rectangular coordinate
system, we identify a vector u with a point in ℝ3 by positioning the initial point of u at the origin,
and defining the components of u to be the coordinates (𝑢1 , 𝑢2 , 𝑢3 ) of its terminal point. To avoid
confusing a vector u with the point (𝑢1 , 𝑢2 , 𝑢3 ), we denote its components by ⟨𝑢1 , 𝑢2 , 𝑢3 ⟩.
If two vectors are equivalent, then they have the same components. Although they may have
different initial points and terminal points, the difference between the coordinates of the terminal
point and those of the initial point are always equal to these components. That is, if a vector v
has initial point 𝑃1 = (𝑥1 , 𝑦1 , 𝑧1 ) and terminal point 𝑃2 = (𝑥2 , 𝑦2 , 𝑧2 ), then the components of the
vector v are
v = ⟨𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 , 𝑧2 − 𝑧1 ⟩.
If the initial point is the origin 𝑂 = (0, 0, 0), we refer to the vector as the position vector of the
terminal point, since the components of the vector are the same as the coordinates of the terminal
point.
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Figure 3: A vector u and various scalar multiples.
Example Let u be a vector with initial point 𝑈1 = (0, 0, 0) and terminal point 𝑈2 = (2, 3, 4). Let
v be a vector with initial point 𝑉1 = (1, 2, −1) and terminal point 𝑉2 = (3, 6, 3). These vectors are
equivalent, and both have components ⟨2, 3, 4⟩. □
The magnitude, or length, of a vector v = ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩, denoted by ∣v∣, is the distance between
its initial and terminal points. It follows from the definiton of the vector’s components that
√
∣v∣ = 𝑣12 + 𝑣22 + 𝑣32 .
Example Let v be a vector with initial point 𝑃1 = (−3, 4, −1) and terminal point 𝑃2 = (1, 5, 0).
Then
√
√
∣v∣ = (1 − (−3))2 + (5 − 4)2 + (0 − (−1))2 = 18 ≈ 4.24.
□
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Summary
∙ A vector is a quantity that has both a magnitude and a direction. It can be represented,
geometrically, by an arrow that points in the vector’s direction, and has a length equal to the
vector’s magnitude. The tip of the arrow is called the vector’s terminal point, and the tail of
the arrow is its initial point.
∙ The zero vector is a vector whose magnitude is zero. It is the only vector that has no specific
direction.
∙ Geometrically, the sum of two vectors u and v is the vector u + v obtained by placing the
terminal point of u at the initial point of v, and defining the sum to be the vector with u’s
initial point and v’s terminal point. This is called the Triangle Law.
∙ Vector addition is commutative: u + v = v + u. This is called the Parallelogram Law.
∙ Given a number 𝑠 and vector u, the vector 𝑠u is a scalar multiple of u and 𝑠 is called a scalar.
It has the same direction as u, and its magnitude is 𝑠 times the magnitude of u.
∙ The negative of a vector u, denoted by −u, is defined by (−1)u. It has the opposite direction
of u, and the same magnitude.
∙ The difference of two vectors u and v, denoted by u − v, is defined to be u + (−v).
∙ The components 𝑢1 , 𝑢2 and 𝑢3 of a vector u = ⟨𝑢1 , 𝑢2 , 𝑢3 ⟩ are defined to be the rectangular
coordinates of its terminal point (𝑢1 , 𝑢2 , 𝑢3 ), when its initial point is the origin.
∙ The magnitude of a vector v√
= ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩ is equal to the distance between its initial point
and terminal point, which is 𝑣12 + 𝑣22 + 𝑣32 .
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