Download 3D Vectors components lecture notes and more Lecture notes Mathematics in PDF only on Docsity!
SESSION 3: COMPONENTS, 3D VECTORS, UNIT VECTORS
Components of a vector given magnitude and direction
Given a vector 𝑨
with magnitude |𝑨
| and direction 𝜽 from
the positive x-axis to the positive y-axis (Illustration 1), we
calculate its x-component 𝑨 𝒙
and its y-component 𝑨
𝒚
by:
Magnitude and direction of a vector given its components
Given the x-component 𝐴
and y-component 𝐴
of vector 𝑨
, we calculate its magnitude |𝑨
| and
direction 𝜃 by:
1
1
tan 𝜃 =
⇒ 𝜃 = arctan <
Resultant Vector
When multiple vectors 𝑨
1
@
B
are added, a resultant vector 𝑹
is generated. Illustration 2 shows
how multiple vectors may be added:
To calculate the x-component 𝑅
and y-component 𝑅
of the resultant vector 𝑹
, we have:
E
1
E
@
E
B
E
G
1 G
@ G
B G
We calculate the magnitude and the direction of vector 𝑹
in the same manner as we do with other vectors.
= 𝐴 cos 𝜃
= 𝐴 sin 𝜃
Illustration
Illustration 2
Vectors in Three Dimensions
A vector 𝑨
in three dimensions is described by its x-, y-, and z-components, K𝐴
L
M (Illustration 3).
The distance between vectors 𝑨
and 𝑩
in three dimensions (Illustration 4) can be calculated by
1
1
L
L
1
Unit Vectors
A unit vector has a magnitude of 1 and is used to describe a direction in space. Vectors may be expressed
in unit-vector notation.
A unit vector 𝚤̂ points in the direction of the positive x-axis, 𝚥̂ in the direction of the positive y-axis, and 𝑘
T
in the direction of the positive z-axis (Illustration 5 and Illustration 6).
Vector 𝑨
with x- and y-components 𝐴
and 𝐴
respectively may be written as:
Illustration 3
Illustration 4
Illustration 5
Illustration 6
(c) To solve for the magnitude d𝐴
d and direction 𝜃 of the resultant vector,
d𝐴
d =
1
1
e( 5 𝑚
1
1
𝜃 = tan
f>
= tan
f>
= tan
f>
Vector Magnitude Direction x-component y-component
""⃗ 10 m 90° 𝟎𝒎 𝟏𝟎𝒎
"⃗ 5 m 0° 𝟓𝒎 𝟎𝒎
"""⃗ 4 m 90° 𝟎𝒎 𝟒𝒎
Resultant
vector 𝑨
15 m 𝟕𝟎° 𝟓𝒎 𝟏𝟒𝒎
(d) To express the resultant vector into unit vector notation,
- Given the vectors below, calculate the distance between 𝐴
and 𝐵
and between 𝐵
and 𝐶
= [ 12 𝑚, 4. 0 𝑚, − 2. 0 𝑚]
= [− 5. 0 𝑚, − 1. 0 𝑚, 6. 0 𝑚]
= [ 10 𝑚, 7. 0 𝑚, 3. 0 𝑚]
(a) To find the distance 𝐴𝐵 between 𝐴
and 𝐵
1
1
L
L
1
1
1
1
e 378 𝑚
1
(b) To find the distance BC between 𝐵
and 𝐶
1
1
L
L
1
e( − 5 𝑚 − 10 𝑚
1
1
1
- Given the following vectors 𝐴
and 𝐵
T
and 𝐵
T
, find 𝐴
and
its magnitude, and 𝐴
and its magnitude.
(a) To find 𝐴
L
L
T
T
= −r
u
(b) To find the magnitude of 𝐴
d𝐴
d = e(− 1 )
1
1
1
(c) To find 𝐴
L
L
T
T
= 𝟑r̂ + 𝟐ŝ + 𝟒𝒌
u
(a) To find the magnitude of 𝐴
d𝐴
d =
e( 3
1
1
1
