This document discusses vectors, including: - Vectors have both magnitude and direction, unlike scalars which only have magnitude. - Vector addition involves combining the components of two or more vectors to determine the resultant vector. - Scalar and vector multiplication can be used to modify the magnitude and direction of vectors. - Properties of vectors include that they can be added through the commutative and associative laws and that multiplying a vector by a scalar scales the vector's magnitude.

1. V E C T O R S
Putri Yeni Aisyah, S.T., M.T.

2. 1. Key Ideas
 Scalars, such as temperature, have magnitude only. (100𝐶).
Vectors, such as displacement, have both magnitude and
direction (5 m, north)
 The vector connecting the tail of the first to the head of the
second is the vector sum (𝑠)
 The (scalar) components 𝑎𝑥 dan 𝑎𝑦 of any two dimensional
vector 𝑎 along the coordinate axes are found by dropping
perpendicular lines from the ends of 𝑎 onto the coordinate axes.
The components are given :
𝑎𝑥 = 𝑎 cos 𝜃
𝑎𝑦 = 𝑎 sin 𝜃
So, we can find the magnitude and orientation the vector 𝑎 with
𝑎 = 𝑎2
𝑥 + 𝑎2
𝑦
tan 𝜃 =
𝑎𝑦
𝑎𝑥

3. We call AC the vector sum
(or resultant) of the vector
AB dan BC.
𝑠 = 𝑎 + 𝑏
Adding
Vectors
.
A vector that has a magnitude
of exactly 1 and points in a
particular directions
𝑎 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗
Unit Vectors
The vectors 𝑏 and −𝑏 have the
same magnitude and opposite
directions
𝑏 + −𝑏 = 0
REMEMBER THAT : We can add/ subtract only vectors of the same
kind.
For ex, we can add two displacements or two velocities, but adding a
disolacement and a velocity makes no sense
𝑖 = 1,0,0 = 𝑖 = 1
𝑗 = 0,1,0 = 𝑗 = 1
𝑘 = 0,0,1 = 𝑘 = 1

4. 𝑎 = 𝑎2
𝑥 + 𝑎2
𝑦 == 𝑎′2
𝑥 + 𝑎′2
𝑦
And
𝜃 = 𝜃′ + ∅
The same vector, with the axes of
the coordinate system rotated
through an angle
Commutative law, 𝐴 + 𝐵 = 𝐵 + 𝐴
Asssociative law, 𝐴 + 𝐵 + 𝐶 = 𝐴 + 𝐵 + 𝐶
Vector addition with Unit Vector,
𝑅 = 𝐴 + 𝐵 = 𝑅𝑥𝑖 + 𝑅𝑦𝑗 = 𝐴𝑥 + 𝐵𝑥 𝑖 + 𝐴𝑦 + 𝐵𝑦 𝑗

5. 2. Key Ideas
 The product of a scalar 𝑠 and vector 𝑣 is a new vector whose
magnitude is 𝑠𝑣 and whose direction is the same as that of 𝑣
is a positive and opposite that of 𝑣 is a negative
 The scalar (or dot) product of two vectors 𝑎 and 𝑏 is written
𝑎 . 𝑏 and is the scalar quantity given by
𝑎. 𝑏 = 𝑎𝑏 𝑐𝑜𝑠Φ
In unit vector notation
𝑎. 𝑏 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘 . (𝑏𝑥𝑖 + 𝑏𝑦𝑗 + 𝑏𝑧𝑘)
 The vector (or cross) product of two vector 𝑎 and 𝑏 is
written 𝑎 x 𝑏 and is a vector 𝑐 whose magnitude c is given
by
𝑎𝑥𝑏 = 𝑎𝑏 𝑠𝑖𝑛Φ
Which Φ is the smaller of two angles between 𝑎 and 𝑏
In unit vector
𝑎𝑥𝑏 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘 𝑥(𝑏𝑥𝑖 + 𝑏𝑦𝑗 + 𝑏𝑧𝑘)
Multiplying
Vectors.

6. Properties of
Vectors.

7. SAMPLE PROBLEM
1. Considering various orientation of vector (a and b), respectively 𝑐 = 𝑎 + 𝑏. What are the maximum possible and minimum
possible magnitude for 𝑐 if 𝑎 and 𝑏 are 3 and 4 m
2. Fig. 1 shows the entrance to such a maze and the first two choices we make at the juctions we encounter in moving from point
i to the point c. A hedge maze is a maze formed by tall rows of hedge. After entering , you search for the center point and the
for the exit. . We undergo three displacements as indicated in the overhead view of Fig. 2.
𝑑1 = 6 𝑚; 𝜃1 = 400
𝑑2 = 8 𝑚; 𝜃2 = 300
𝑑3 = 5 𝑚; 𝜃3 = 00
When we reach point c, what are the magnitude and angle of our net displacement 𝑑𝑛𝑒𝑡 from point i
Fig. 1 Fig. 2

8. THANK YOU