Vector quantities always have its direction and magnitude, while scalar quantities are only have its magnitude.
A vector is a quantity that has both magnitude and direction
A. Symbolism of Vectors
The magnitude of vector A expressed as an arrow.
Vector A and B have the same magnitude but differ in direction.
B. Addition of Vectors
When adding vectors, we must also consider the directions of each vector to be summed. That's why we use below methods:
Triangle method
We use parallelogram to draw 2 vectors only.
For the example question, you can check this out!https://www.youtube.com/watch?v=EQ4LFSixuHE
The addition of large numbers of vectors can be performed easily using a polygon method.
Triangle and polygon method application:
https://www.youtube.com/watch?v=gtXZ97uEjkU
C.Subtraction of Vectors
Subtraction of two vectors is identical to the addition of a vector with the negative of the other vector.
Thus, the subtraction is:
C= A-B
D. Components of vector
The magnitude of vector B (projection of vector C along along the Y-axis) is called the component of vector C in the Y-direction (Fy).
Symbol θ represents the angle formed by vector C and the X-axis.
E. Analytical addition
We can perform large numbers of vectors addition more simply using the analytical method.
If there are only 2 vectors, we use this formula:
If there are more than 2 vectors and degrees, we use this formula:
For the example question, check this out!
https://www.youtube.com/watch?v=g_TnqKX5ybY
F. Dot and Cross Product
The dot product of two vectors is the magnitude of one times the projection of the second onto the first. The symbol that used in dot product is (.) Dot formula always use cos.
(x = i,y = j,z = k)
The cross product of two vector is the area of the parallelogram
between them. the symbol that used in cross product is (×). Cross formula always using sin.
B x
A = B (A sin θ) = BA sin θ
The
different between dot and cross product is that the cross product has
its magnitude and direction, while dot product just has its magnitude.
The rule of cross product:
There
are 2 directions: Into the page (-) and out of the page(+). The way to
find direction of cross product is using the right hand rule. Here are
the steps:
1.Hold your right hand flat with your thumb
perpendicular to your fingers. Do not bend your thumb at anytime.
2.
Point your
fingers in the direction of the first vector.
3.
Orient your
palm so that when you fold your fingers they point in the direction of the
second vector.
4.
Your thumb
is now pointing in the direction of the cross product.
Examples:
There are two vectors. A= 8m to the south west (30°to the respect of south), B= 10 m to the east. Determine the result of:
A) A . B
B) B . A
C) A × B
D) B × A
Answer:
A) A . B = (8)(10) cos 120°=-40m
B) B . A = (10) (8) cos 120°=-40m
C) A × B=(8)(10) sin 120°=40√3 m into the page
D) B × A=(10)(8) sin 120°=40√3 m into the page
