Introducing
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In this chapter, vectors are introduced and we learn about concepts, properties, and applications of vectors in two space and three space.
Scalar: Any quantities described only the magnitude. For example, age, height, area and temperature.
Vectors: Any quantities described by the magnitude and direction. For example, velocity, friction and gravity.
Opposite vectors: Vectors that have the same magnitude but opposite direction.
Geometric vectors: These vectors have an initial point and a final point.
Equal vectors: Vectors with the same magnitude as well as the same direction.
For example, a truck is traveling at 60 km/hr to the west. The magnitude is 60 km and the direction is west. This is a vector quantity. This diagram is a geometric vector, which is a representation of a vector using an arrow diagram, or directed line segment, that shows both magnitude (or size) and direction. The length of the arrow represents and is proportional to the vector’s magnitude.
Parallelogram law
triangle law
Definitions
Given two vectors with an angle between them, there are two methods of adding these vectors to get the sum of the two resultant vectors.
Resultant vector: It has the same effect as two original vectors applied on after each other.
Zero: The vector that results from adding two equal and opposite vectors together. It has no real direction.
The triangle law of vector addition asserts that when two vectors are represented as two sides of a triangle in the order of magnitude and direction, the third side of the triangle indicates the magnitude and direction of the resultant vector.
Parallelogram law: It is the law about completing the parallelogram formed after placing two vectors tail to tail to find the resultant vector from adding two vectors together. Vectors a and b can be added from tip to tail to form a resultant vectors denotes as a⃗⃗ + b⃗ . Example of a parallelogram:
Triangle law of vector additional: The idea is to complete the triangle formed by positioning the two vectors head to tail.
Multiplication of a vector by a scalar
This lesson, we learn the effect of multiplying a vector by a scalar k:
- f k>1 the vector increases in magnitude
- if 0>k>1 the vector decreases in magnitude
- if k<0 the vector will change to the opposite direction
- if k is 0, then the result is a zero vector
Unit vectors are any vectors with a length of 1 including basis vectors
Collinear vectors can be translated so they lie on the same line.
Two vectors are collinear if there is a non zero k value.
Dividing a vector by its magnitude will result in a unit vector.
Commutative Property of Addition
a + b = b + a
Associative Property of Addition
(a + b) + c = c + (a + b)
Distributive Property of Addition
k(a + b) = ka + kb
Adding 0
a + 0 = a
Associative Law for Scalars
m(na) = n(ma)
Distributive Law for Scalars
(m + n)a = ma + na
Vectors in R2 and R3
In this lesson, we are introduced to position vectors:
Position vector - |OP| has its tail at the origin and is therefore called a position vector.
in three dimensional space we have a third value in the coordinate (x,y,z). The origin is (0,0,0) and points are written in the form P(x,y,z). There are three coordinate planes known as the xy-,xz-, and yz- planes.
A cartesian plane in R3 has three planes:
the xy-plane, formed by the x and y axes, with OP having coordinates of (a,b,0)
the yz-plane, formed by the y and z axes, with OP having coordinates of (0,b,c)
the xz-plane, formed by the x and z axes, with OP having coordinates of (a,0,c)
Geometric vectors: vectors that are represented as a line segment, proportional to the magnitude of the vector.
Algebraic vectors: Vectors drawn on a coordinate axis represented by their components such as (a,b).
Position vectors: a vector, OP, that has its tail at the origin, O and it’s head at a point, P in R2
Unit vector: vector with a magnitude of 1, unit vectors along the x-axis are represented as f and unit vectors along the y-axis are represented as g, also known as standard unit vectors.
The position vector, OP can be represented as OP= (a,b) or as OP= af+bg
We are using unit vectors i and j, but now in R3 we use k as well. For a vector (x,y,z), it can be written in the form of the unit vectors i, j, and k. (xi, yj, zk)
In R3 ,
the unit vector i= (1,0,0) lies along the x-axis
The unit vector j= (0,1,0) lies along the y-axis
The unit vector k= (0,0,1) lies along the z-axis
Algebraic vectors: vectors drawn on the coordinate axis, represented by their components.
Any vector in R3 can be represented as the sum of its unit vectors.
All properties apply
Addition and subtraction if u= (u1 +u2+u3) and v= (v1+ v2+ v3), the u+v= (u1+v1, u2+v2, u3+v3) and u-v= ( u1-v1, u2-v2, u3-v3).
Vector between two points- P (x1,y1,z1) and Q (x2,y2,z2) is PQ= (x1 +X2, y1+y2, Z1+Z2).
Non-collinear vectors: it is a linear combination of these vectors is au+bv which represents the diagonal formed and a and b are scalars.
We learned that these are spanning sets and spans R2; the two vectors can be written as a linear combination from every vector in the plane.
When two vectors span R2 or a plane in R3,every vector in that plane can be written as a linear combination of the original vectors.
Spanning sets: if the vectors u and v are non collinear vectors, then the combination of this set of vectors spans R2 This is considered a spanning set. This indicates that every vector in R2 can be written uniquely as a linear combination of the unit vector.
Force is determined by multiplying mass (kg) and acceleration (m/s2). The resulting unit is in Newtons (N). Because force is calculated by multiplying a scalar (mass) and vector (acceleration), force itself is a vector.
If an object has multiple forces acting on it, or multiple vectors, the single force that is used to represent the combination of these forces (components) is the resultant force
Equilibration Forces: A number of vector forces that oppose when acting on an object. This force maintains the object in a state of equilibrium
Resultant forces: the single force of multiple forces acting on an object
Resolution: is splitting apart forces into its components, the most useful way you could do this is by having these components make a 90 degrees
For example,
Determine the equilibrant of two forces of 12N and 20 N acting at an angle of 75 degrees to each other.
r^2 = 12^2 + 20^2-2(12)(20) cos105
r= 25.9 N
Sin A/20= Sin 105/25.9
A= 48 degrees.
Therefore, the equilibrant is 25.9 N in magnitude 27 degrees from the 20 N towards the 12 N.
The lesson we learned the concept of velocity as a vector quantity and using addition of vectors to get resultant velocity.
Air velocity: as an object travels, the velocity of air flow will create resultant velocity and change the direction of the object
Ground velocity: as an object travels, the velocity of water flow will create resultant velocity or change the direction of the object.
Dot product: When we multiply two vectors, it’s called the dot product. The answer will always be in scalar quantity. One application is Work which is force multiplied by distance.
The dot product of two geometric vectors is given by (a)(b)= |a||b|cos x
Where x is the angle between the two tails of a and b
Example: |u| = 4, |v|= 5 and x= 10
uv= (4)(5)cos10
uv= 9.4
Dot product for algebraic vectors
The scalar projection is a scalar, equal to the length of the orthogonal projection of on , with a negative sign if the projection has an opposite direction with respect to
Vector projection: To find the vector(magnitude and direction), we only have to multiply the scalar projection by the unit vector in the direction of the vector onto which the other one is being projected.
Direction of cosine and direction angles
Direction angle: the angles that a position vector makes with the x,y and z-axis. To calculate it, you need to project the direction vector onto each of the axis using the unit vector that belongs to that axis.
To calculate, we would need to project the direction vector onto each of the axis using the unit vectors belonging to that axis.
This lesson introduces the cross product which is a vector that is perpendicular to both vectors and therefore is only possible in R3
If you find the dot product of two vectors, you get a scalar product. However, if you find the cross product of two vectors, you get a vector.
Application of Dots and cross product
Using the dot product, we learned about Work (j) which is scalar and measured in joules. It is calculated by taking the dot product of the force (N) and the displacement (m) vectors of an object.
Torque: It is a perpendicular force applied onto an object from the angular force (N) and radius (m) of the object. It is calculated by cross product.
Area of parallelogram: It is given by multiplying the base times the height. The height is perpendicular to the base using the cross product. We can also find the area of a triangle by dividing it by two
To calculate rational force, we use cross product to find torque using joules as well, using radius and torque.
This chapter introduces vector, parametric and symmetric equations of a line and plane as well as using Cartesian equations.
vector equation
parametric equations of a line in r^2
Two lines are parallel if they are scalar multiples of each other, which means their normals and direction vectors would be parallel and scalar multiples of each other.
Two lines are perpendicular if their dot product is zero. The dot product of their direction vectors would also be zero.
Cartesian equation: Cartesian equation of a line: Ax + By + C = 0, the normal to this line is n = (A,B)
The Cartesian equation of a plane follow the same principles as the Cartesian equation of line; Ax+ By+ Cz + D = 0 where n= (a,b,c) is a normal vector to this plane.