MATERI KELAS X

Chapter 3: Vectors

• To describe motions in 2- or 3-dimensions, we
need vectors
•A vector quantity has both a magnitude and a
direction. e.g., acceleration, velocity, displacement,
force, torque, and momentum.
•A scalar quantity does not involve a (spatial)
direction. e. g. charge, mass, time, temperature,
energy, etc.Space
We need three spatial dimensions to describe a definite
location in space. In the Cartesian (rectangular)
coordinate system, the dimensions are labeled: “x”, “y”,
and “z”.
To describe 3-D motion, we will dissect it into 3
one-dimensional motions. Each 1-D will be a
component in the larger 3-D vector.
All of the concepts from Chapter 2 *will* apply to
each component (x, y, z) in 3-D.Unit Vectors
• Has a magnitude of 1 and points in a
particular direction. ONLY indicates
DIRECTION!

• Graphical method
e. g. two vectors a, b
a + b =
a – b = a + (- b) =
The negative of a vector turns it 180o.• Check point 3.1: The magnitude of
displacements a and b are 3 m and 4 m,
respectively, and c = a + b. Considering
various orientations of a and b, what is
(a) the maximum possible magnitude for c
(b) the minimum possible magnitude for cVector Addition Property
a + b = b + a
(commutative)
(a + b) + c = a + (b + c)
(associative)Vector in a coordinate system

θComponents of Vectors
• Component notation vs

=Add vectors by components

In which quadrant would a + b
be located if
a = 3.0 i − 4.0 j and
b = −2.0 i + 2.0 j?Daily Quiz, August 27, 2004

In which quadrant would a + b
be located if
a = 3.0 i − 4.0 j and
b = −2.0 i + 2.0 j?Multiplication of Vectors
• Multiply a vector by a scalar: b = s a
– Magnitude of b: s times the magnitude of a
– Direction of b : same as a if s > 0,
opposite of a if s < b =" a" b =" a" 0o =""> a.b = a b cos 0o = a b
if a and b perpendicular, φ = 90o
=> a.b = a b cos 90o = 0
i.i = j.j = k.k = 1 i.j = i.k = j.k = 0Scalar Product

rCheck point 3-4: Vectors C and D have magnitudes
of 3 units and 4 units, respectively. What is the
angle between the directions of C and D if C.D
is
(a) zero
(b) 12 units
(c) –12 unitsScalar Product between a and b

Note: a x b = - (b x a)Vector product
• c = a x b => c = a b sinφ
•if a and b parallel, φ = 0o
=> a x b = 0
if a and b perpendicular, φ = 90o
=> c = a b
i x i = j x j = k x k = 0
i x j = j x k = k x i = 1
j x i = k x j = i x k = −1Check point 3-5: Vectors C and D have magnitudes
of 3 units and 4 units, respectively. What is the
angle between the directions of C and D if the
magnitude of the vector products C x D is
(a) Zero?
(b) 12 units?