Updated: January 10, 2016
Calculus III
Section 12.2
Math 232
Calculus III
Brian Veitch •
Fall 2015 •
Northern Illinois University
12.2 Vectors
Definition 1: Vector
A vector is a quantity that has both direction and magnitude (length). We write
~
~v = AB
Definition 2: Vector Addition
~ and ~v = BC
~
Geometrically: Let ~u = AB
~ + BC
~ = AC
~
AB
~ = ~u + ~v . This is called the Head to Tail method of Vector Addition.
where AC
Vectors Sharing the Same Initial Point
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Calculus III
Section 12.2
To show why ~u +~v is the vector shown above we need to first understand a vector a little
more. Because a vector is defined to be simply length and direction, any vector with those
two properties are considered equivalent. Each vector in the graph below are considered the
same vector ~u.
Position Vector: The position vector is always the vector that has its initial point at the
origin (0, 0).
Let’s get back to vector addition. If both vectors share the same initial point you can
move one of the vectors (making sure to keep the same direction and length) so that the tail
of ~v starts at the head of ~u. Then using the originally defined vector addition from above
we get
You can also see that it forms a parallelogram. This is known as the Parallelogram Law.
This also shows that
~u + ~v = ~v + ~u
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Calculus III
Section 12.2
Definition 3: Vector Subtraction
Let ~u and ~v be vectors. Then ~u − ~v = ~u + (−~v )
What’s −~v ? You rotate the vector ~u 180 degrees so that it’s pointing in the opposition
direction.
Again given vectors ~u and ~v , then ~u − ~v is shown below
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Calculus III
Section 12.2
Definition 4: Scalar Multiplication
Multiplying a vector ~v by a scalar c will stretch or compress the vector.
If c > 0 the vector c~v keeps the same direction.
If c < 0 the vector c~v points in the opposite direction.
Example 1
If vectors ~u and ~v are defined below, find the vectors 2~u, − 13 ~v , ~u +~v , ~v −~u, and 2~u − 3~v
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Calculus III
Section 12.2
Definition 5: Vectors Componenets
Let ~a be a vector defined by ~a =< a1 , a2 > or ~a =< a1 , a2 , a3 >. a1 , a2 and a3 are
called the components of vector ~a.
The components are the displacement from the initial point to its terminal.
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Calculus III
Definition 6: Creating a Vector from Two Points
Definition 7: Vector Magnitude (length)
p
a21 + a22
p
Let ~a =< a1 , a2 , a3 >, then the magnitude is |~a| = a21 + a22 + a23
Let ~a =< a1 , a2 >, then the magnitude is |~a| =
Definition 8: Vector Addition/Subtraction, Scalar Multiplication
For 2D: Let ~a =< a1 , a2 >, ~b =< b1 , b2 >, and c be a scalar, then
~a + ~b =< a1 + b1 , a2 + b2 >
~a − ~b =< a1 − b1 , a2 − b2 >
c~a =< ca1 , ca2 >
For 3D: Let ~a =< a1 , a2 , a3 >, ~b =< b1 , b2 , b3 >, and c be a scalar, then
~a + ~b =< a1 + b1 , a2 + b2 , a3 + b3 >
~a − ~b =< a1 − b1 , a2 − b2 , a3 − b3 >
c~a =< ca1 , ca2 , ca3 >
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Section 12.2
Updated: January 10, 2016
Calculus III
Section 12.2
Example 2
Let ~a =< 4, 0, 3 > and ~b =< −2, 1, 5 >. Find ~a + ~b, ~a − ~b, 3~b, |~a|, |~b − ~a|, and 2~a + 4~b
1. ~a + ~b =< 4, 0, 3 > + < −2, 1, 5 >=< 2, 1, 7 >
2. ~a − ~b =< 4, 0, 3 > − < −2, 1, 5 >=< 6, −1, −2 >
3. 3~b = 3 < −2, 1, 5 >=< −6, −3, 15 >
4. |~a| =
p
√
(4)2 + (0)2 + (3)2 = 25 = 5
5. |~b − ~a| = | < −6, 1, 2 > | =
p
(−6)2 + (1)2 + (2)2 =
√
36 + 1 + 4 =
√
41
6. 2~a + 5~b = 2 < 4, 0, 3 > +5 < −2, 1, 5 >=< 8, 0, 6 > + < −10, 5, 25 >=< −2, 5, 31 >
Note that all arithmetic properties are true. For example, ~a + ~b = ~b + ~a
Definition 9: Standard Basis Vectors (Unit Vectors)
i =< 1, 0, 0 >, j =< 0, 1, 0 >, k =< 0, 0, 1 >
This means the vector ~a =< a1 , a2 , a3 > can be written as ~a = a1 i + a2 j + a3 k
For example, ~a =< −1, 4, 6 >= −i + 4j + 6k or ~b =< 4, 2 >= 4i + 2j.
~a + ~b = (−i + 4j + 6k) + (4i + 2j + 0k) = 3i + 6j + 6k =< 3, 6, 6 >
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Calculus III
Section 12.2
Definition 10: Unit Vector
A unit vector is a vector with length 1. If ~a is any vector, then
~a
is a unit vector
|~a|
To find a vector with the direction of ~a with length L
~v =
L
~a
|~a|
Example 3
Let ~a =< 2, −1, −2 >. Find the unit vector with the direction of ~a.
1. First, find |~a|.
|~a| =
p
(2)2 + (−1)2 + (−2)2 =
√
9=3
2. To create a unit vector, divide ~a by 3.
~u =
1
2 1 2
~a
= < 2, −1, −2 >=< , − , − >
|~a|
3
3 3 3
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