Vectors in the TwoDimensional Space

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Vectors in the Two-Dimensional Space
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Definition of Vectors
Quantities that are determined by a magnitude
alone are called scalars.
e.g. temperature, density, length, speed, distance
Quantities that have both magnitude and direction
are called vectors.
e.g. displacement, velocity, acceleration, force
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(No Transcript)
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A vector may be represented geometrically by a
directed line segment or an arrow whose direction
represents the direction of the vector and whose
length represents the magnitude.
The vector is denoted by
a
and its magnitude is denoted by
a
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Two vectors are said to be equal if and only if
they have the same magnitude and direction.
ABCD is a parallelogram,
then and
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A vector with zero magnitude is called a zero
vector. Therefore is always a zero
vector. A zero vector is denoted by 0. It is
the only vector that has no specific direction.
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Unit Vector
A vector with magnitude 1 unit is called a unit
vector. That is, u is a unit vector if and only
if u 1.
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The negative vector of v, denoted by v, is a
vector having equal magnitude but opposite
direction to v. Therefore
and .
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Basic Operation of Vectors
(A) Addition
C
triangle law of addition
A
B
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Basic Operation of Vectors
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Basic Operation of Vectors
parallelogram law of addition
B
C
O
A
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(B) Subtraction
B
b
a - b
a b a (-b)
O
A
a
a (-b)
-b
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(C) Scalar Multiplication
The product of a vector a and a scalar k is a
vector, denoted by ka. This operation is called
scalar multiplication.
If k 3.
a
3a
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Laws of Vectors

• u v v u

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Laws of Vectors

• u v v u
• u (v w) (u v) w

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Laws of Vectors

• u v v u
• u (v w) (u v) w
• m(nu) (mn)u n(mu)

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Laws of Vectors

• u v v u
• u (v w) (u v) w
• m(nu) (mn)u n(mu)
• (m n)u mu nu

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Laws of Vectors

• u v v u
• u (v w) (u v) w
• m(nu) (mn)u n(mu)
• (m n)u mu nu
• m(u v) mu mv

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Exercise 11.2 P.155
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Vectors in the Rectangular Coordinates System
OMPN is a rectangle,
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xi yj
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x1i y1j
x2i y2j
(x2i y2j) - (x1i y1j)
(x2 - x1)i (y2 - y1)j
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Its magnitude and direction are given by
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Properties of Component Vectors
ai bj ci dj if and only if
a c and b d
ai bj 0 if and only if a
b 0
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Point of Division
(A) Position Vector
B
b
A
a
O
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Usually, the reference point O is chosen to be
the origin of the xy-plane. Thus position vectors
can be expressed in terms in terms of i and j.
Therefore any vector on a plane can be expressed
in terms of position vectors.
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(B) Point of Division
Let A and B be two points with position vectors a
and b with respect to O. If P is a point on the
line AB such that
n
B
P
m
A
b
p
a
O
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Then the position vector of is given by
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If P is the mid-point of AB, then m n 1. The
formula becomes
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Exercise 11.3 P.162
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Applications of Vectors (I) ---- Prove the lines
are parallel by vectors

• Given two non-zero vectors u and v and a real
  number k. If u kv, then u and v are parallel.

If two non-zero vectors u and v are parallel,
then u kv, where k is a non-zero real number.
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Applications of Vectors (II) ---- Prove three
lines are collinear by vectors
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Applications of Vectors (III) ---- Use vectors to
find the ratio of line segments on a straight
line.

• Given two non-zero vectors a and b which are
  not parallel, k1, k2, m1, m2, n1 and n2 are real
  numbers. We have following properties

• If k1a k2b 0, then k1 k2 0.
• If m1a n1b m2a n2b, then m1 m2 and n1
  n2.
• If a vector u m1a n1b is parallel to v m2a
  n2b, then , where m2 and n2
  are non-zero.

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Exercise 11.4 P.170
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Scalar Product (Dot Product)
The scalar product of two vectors a and b,
denoted by (read as a dot b), defined as
the product of magnitudes of a and b and the
cosine of the angle ? (where
) between them. In symbolic form,
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Do not write as
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Cases of dot product

• (1) angle between the 2 vectors is acute

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Cases of dot product

• (2) angle between the 2 vectors is right-angled

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Cases of dot product

• (3) angle between the 2 vectors is obtuse

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Laws of dot product

• 1.
• 2.
• 3.
• 4.

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Laws of dot product

• 5.
• 6.
• 7.

(a and b are both non-zero vectors)
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If a x1i y1j, and b x2i y2j
a . b (x1i y1j) .(x2iy2j)
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If a x1i y1j, and b x2i y2j
a . b (x1i y1j) .(x2iy2j) x1x2i . i
x1y2i . j x2y1j . i y1y2j . j
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If a x1i y1j, and b x2i y2j
a . b (x1i y1j) .(x2iy2j) x1x2i . i
x1y2i . j x2y1j . i y1y2j . j x1x2(1)
x1y2(0) x2y1(0) y1y2(1)
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If a x1i y1j, and b x2i y2j
a . b (x1i y1j) .(x2iy2j) x1x2i . i
x1y2i . j x2y1j . i y1y2j . j x1x2(1)
x1y2(0) x2y1(0) y1y2(1) x1x2 y1y2
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If a x1i y1j, and b x2i y2j
a . b (x1i y1j) .(x2iy2j) x1x2i . i
x1y2i . j x2y1j . i y1y2j . j x1x2(1)
x1y2(0) x2y1(0) y1y2(1) x1x2 y1y2
a . b x1x2 y1y2
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In general, the angle ? between a x1i y1j,
and b x2i y2j
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Exercise 11.5 P.179
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Applications of Vectors (IV) ---- Prove two lines
are perpendicular by vectors
If the scalar product of two non-zero vectors is
zero, then the two vectors are perpendicular to
each other. We can use this fact to prove two
lines are perpendicular to each other.
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Applications of Vectors (V) ---- Solve problems
in coordinate geometry and trigonometry by vectors
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Exercise 11.6 P.185