13.2 Vectors
1. Definition (Vector) In nature, there are objects that have both magnitude and direction such as wind, stream, and
various forces. To visualize such objects, we use a convention called vector. A vector is a geometric object
which consists of length(magnitude) and direction.
We call the starting point and the ending point of a vector the
initial point and the terminal point respectively. When A is the
initial point and B is the terminal point of a vector, we write the
−−→
vector AB.
−−→
(e.g.) Let A(1, 2, 0) and B(3, −1, 2) be two points in rectangular coordinates. Draw AB and find the length
of the vector.
2. Remarks
(a) We call a vector whose length is 1 a unit vector.
(b) A vector is determined by the length and the direction only. Vectors who have the same lengths and
the same directions are therefore considered as the same vectors regardless of the initial point.
3. Definition (Vector represented by a point) When a vector is represented by the coordinates of a point, < a, b, c >, we
assume that (0, 0, 0) is the initial point and (a, b, c) is the terminal point of the vector.
4. Definition (Length, addition, scalar multiplication and linear combination of vectors)
Suppose u =< a1 , b1 , c1 > and v =< a2 , b2 , c2 > are vectors in R3 and α is a constant.
(a) The length(or magnitude) of u, denoted by kuk (or |u|):
kuk =
p
a21 + b21 + c21
(b) The addition/subtraction of vectors:
u ± v =< a1 , b1 , c1 > ± < a2 , b2 , c2 >=< a1 ± a2 , b1 ± b2 , c1 ± c2 >
(c) Scalar multiplication of a vector:
αu = α < a1 , b1 , c1 >=< αa1 , αb1 , αc1 >
(d) Linear combination of vectors:
When v1 , v2 , · · · vk are vectors and α1 , · · · αk are constants, we call the form
1
α1 v1 + α2 v2 + · · · + αk vk
a linear combination of the vectors v1 , · · · , vk .
5. Example Let a =< 1, −2, −3 > and b =< −3, 1, 0 >. Draw a, b and the given linear combination of the vectors,
and then find the length of the vector.
(a) a + b
(b) 3a − 2b
−−→
6. Property If P (a1 , b1 , c1 ) and Q(a2 , b2 , c2 ) are given, then the vector P Q can be represented by a point as
−−→ −−→ −−→
P Q = OQ − OP =< a2 − a1 , b2 − b1 , c2 − c1 >
(e.g.) Let A(1, 3, −2), B(0, 0, 0), C(a1 , b1 , c1 ) and D(a2 , b2 , c2 ) be points in rectangular coordinates. Represent the given vector by a point.
−−→
(a) AB
−−→
(b) BC
−−→
(c) CD
−−→
(d) DA
7. Definition (Standard basis vectors) We use the notation
i =< 1, 0, 0 >, j =< 0, 1, 0 >, k =< 0, 0, 1 >
and call these three vectors the standard basis vectors.
8. Remark Notice that any vector in R3 can be written in a linear combination of the standard basis vectors.
(e.g) Write the vector as a linear combination of the standard basis vectors.
9. Example
(a) < 3, 2, 1 >
(b) < −3, 0, 5 >
(c) < a, b, c > (a, b, c are constants.)
1
(a) Find 3i − j + k
.
2
(b) Find the unit vector in the direction of i + j + k.
2