Grade 10 - Mechanics - LO.4 - Vectors

 

   

We have in LO.4 Mechanics G10

First: the Concepts

A. Vector and scalar quantities

B. Rectangular coordinates

C. Polar coordinates,

D. Engineering (i,j,k)notation

E. Physical diagrams

F. Position vectors

G. Resultant of two vectors

H. Relative velocity in 2-dimensions

Second: the References

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Third: the Videos links

Fourth: Skills

A. Represent a vector using rectangular coordinates, polar coordinates, or engineering (i,j,k)notation

B. Discriminate between vector quantities(e.g. displacement, velocity, force, acceleration) and scalar quantities that have magnitude but not direction (e.g. distance, speed)

C. Translate fluently among different representations of a vector, including: rectangular coordinates, or engineering (i,j,k) notation, polar coordinates, physical diagrams, verbal descriptions, position vectors.

D. Compute relative velocities in classical mechanics (i.e., with speeds substantially less than the speed of light) in 2D

Fifth: the materials as PPT., DOCX., and PDF

In the Drive from this link

Few Notes:

Vectors

Vectors are geometrical entities that have magnitude and direction. A vector can be represented by a line with an arrow pointing towards its direction and its length represents the magnitude of the vector. Therefore, vectors are represented by arrows, they have initial points and terminal points. The concept of vectors was evolved over a period of 200 years. Vectors are used to represent physical quantities such as displacement, velocity, acceleration, etc.

Further, the use of vectors started in the late 19th century with the advent of the field of electromagnetic induction. Here, we will study the definition of vectors along with properties of vectors, formulas of vectors, operation of vectors along using solved examples for a better understanding.

What are Vectors?

A vector is a Latin word that means carrier. Vectors carry a point A to point B. The length of the line between the two points A and B is called the magnitude of the vector and the direction of the displacement of point A to point B is called the direction of the vector AB. Vectors are also called Euclidean vectors or Spatial vectors. Vectors have many applications in maths, physics, engineering, and various other fields.

Vectors in Euclidean Geometry- Definition

Vectors in math is a geometric entity that has both magnitude and direction. Vectors have an initial point at the point where they start and a terminal point that tells the final position of the point. Various operations can be applied to vectors such as addition, subtraction, and multiplication. We will study the operations on vectors in detail in this article.

Vectors - Examples

Vectors play an important role in physics. For example, velocity, displacement, acceleration, force are all vector quantities that have a magnitude as well as a direction.

Representation of Vectors

Vectors are usually represented in bold lowercase such as a or using an arrow over the letter as $\overrightarrow{a}$. Vectors can also be denoted by their initial and terminal points with an arrow above them, for example, vector AB can be denoted as $\overrightarrow{AB}$. The standard form of representation of a vector is $\overrightarrow{A}=a\hat{i}+b\hat{j}+c\hat{k}$. Here, a,b,c are real numbers and $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the x-axis, y-axis, and z-axis respectively.

The initial point of a vector is also called the tail whereas the terminal point is called the head. Vectors describe the movement of an object from one place to another. In the cartesian coordinate system, vectors can be denoted by ordered pairs. Similarly, vectors in 'n' dimensions can be denoted by an 'n' tuple. Vectors are also identified with a tuple of components which are the scalar coefficients for a set of basis vectors. The basis vectors are denoted as: e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1)

Magnitude of Vectors

The magnitude of a vector can be calculated by taking the square root of the sum of the squares of its components. If (x,y,z) are the components of a vector A, then the magnitude formula of A is given by,

|A| = √ (x²+y²+z²)

The magnitude of a vector is a scalar value.

Angle Between Two Vectors

The angle between two vectors can be calculated using the dot product formula. Let us consider two vectors a and b and the angle between them to be θ. Then, the dot product of two vectors is given by a·b = |a||b| cosθ. We need to determine the value of the angle θ. The angle between two vectors also indicates the directions of the two vectors. θ can be evaluated using the following formula:

θ = cos^-1[(a·b)/|a||b|]

Types of Vectors

The vectors are termed as different types based on their magnitude, direction, and their relationship with other vectors. Let us explore a few types of vectors and their properties:

Zero Vectors

Vectors that have 0 magnitude are called zero vectors, denoted by $\overrightarrow{0}$ = (0,0,0). The zero vector has zero magnitudes and no direction. It is also called the additive identity of vectors.

Unit Vectors

Vectors that have magnitude equals to 1 are called unit vectors, denoted by $\hat{a}$. It is also called the multiplicative identity of vectors. The magnitude of a unit vectors is 1. It is generally used to denote the direction of a vector.

Position Vectors

Position vectors are used to determine the position and direction of movement of the vectors in a three-dimensional space. The magnitude and direction of position vectors can be changed relative to other bodies. It is also called the location vector.

Equal Vectors

Two or more vectors are said to be equal if their corresponding components are equal. Equal vectors have the same magnitude as well as direction. They may have different initial and terminal points but the magnitude and direction must be equal.

Negative Vector

A vector is said to be the negative of another vector if they have the same magnitudes but opposite directions. If vectors A and B have equal magnitude but opposite directions, then vector A is said to be the negative of vector B or vice versa.

Parallel Vectors

Two or more vectors are said to be parallel vectors if they have the same direction but not necessarily the same magnitude. The angles of the direction of parallel vectors differ by zero degrees. The vectors whose angle of direction differs by 180 degrees are called antiparallel vectors, that is, antiparallel vectors have opposite directions.

Orthogonal Vectors

Two or more vectors in space are said to be orthogonal if the angle between them is 90 degrees. In other words, the dot product of orthogonal vectors is always 0. a·b = |a|·|b|cos90° = 0.

Co-initial Vectors

Vectors that have the same initial point are called co-initial vectors.

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