Download Vectors Geometrically, we interpret vectors as directed line

Vectors
Geometrically, we interpret vectors as directed line
segments. If P = (x1 , y1 , z1 ) and Q = (x2 , y2 , z2 ),
then
−−→
P Q = (x2 − x1 , y2 − y1 , z2 − z1 ),
where x2 −x1 , y2 −y1 , and z2 −z1 are the components
−−→
−−→
of P Q. The length of the vector P Q is
p
−−→
|P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
z
5
Q
4
P
3
2
1
1
O
1
2
2
3
4
5
x
1
3
4
5
y
Example. Find the vector represented by the directed line segment with initial point A(2, −3, 4) and
terminal point B(−2, 1, 1). Moreover, find the length
of the vector.
Solution. We have
−−→
AB = (−2 − 2, 1 − (−3), 1 − 4) = (−4, 4, −3)
and
p
√
−−→
2
2
2
|AB| = (−4) + 4 + (−3) = 41.
Two vectors are considered to be equal if these
two vectors are parallel to each other and have the
same direction and the same length.
2
Addition and Subtraction
If u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) are two
vectors, then
u + v = (u1 + v1 , u2 + v2 , u3 + v3 )
u − v = (u1 − v1 , u2 − v2 , u3 − v3 ).
In the following parallelogram ABCD, we have
−−→ −−→
−−→ −−→
AB = DC and AD = BC.
Moreover,
−→ −−→ −−→
−−→ −→ −−→
AC = AB + AD and BC = AC − AB.
D
C
A
B
3
Scalar Multiplication
If v = (v1 , v2 , v3 ) is a vector and c is a real
number, then
cv = (cv1 , cv2 , cv3 ).
If c > 0, cv and v have the same direction. If c < 0,
cv and v have opposite directions.
Let u be a vector and v a nonzero vector. Then
u is parallel to v if and only if there exists a real
number t such that u = t v.
3v
v
-2v
v
4
Algebraic Operations of Vectors
Let u, v, and w be three vectors, and let b and
c be two real numbers. Then we have
u + v = v + u,
(u + v) + w = u + (v + w),
u + 0 = u,
u + (−u) = 0,
b(u + v) = bu + bv,
(b + c)u = bu + cu,
b(cu) = (bc)u,
1u = u.
Example. Let u = (4, 0, 3) and v = (−2, 1, 5). Find
u + v, u − v, 3v and 2u + 5v.
Solution. We have
u + v = (2, 1, 8),
u − v = (6, −1, −2),
3v = (−6, 3, 15),
and
2u + 5v = (8, 0, 6) + (−10, 5, 25) = (−2, 5, 31).
5
Unit Vectors
Recall that the length of a vector u = (u1 , u2 , u3 )
is given by
p
|u| = u21 + u22 + u23 .
For a real number c, we have |cu| = |c||u|.
A unit vector is a vector whose length is 1.
The three vectors
i = (1, 0, 0),
j = (0, 1, 0),
k = (0, 0, 1)
are called unit coordinate vectors. Any vector
u = (u1 , u2 , u3 ) can be expressed as
u = u1 i + u2 j + u3 k.
6
If a is a nonzero vector, then the unit vector in
the direction of a is given by
a
u=
.
|a|
Example. Find the unit vector in the direction of
the vector a = 2i − j − 2k.
Solution. We have
p
|a| = 22 + (−1)2 + (−2)2 = 3.
The unit vector in the direction of a is
a
2
1
2
u=
= i − j − k.
|a|
3
3
3
7