About vector/complex number

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In summary, a vector is a mathematical object that represents magnitude and direction and is commonly used in physics, engineering, and mathematics. Vectors can be represented in different ways such as coordinates, components, or unit vectors. Operations such as addition, subtraction, and multiplication can be performed on vectors to solve problems. Complex numbers, which have both a real and imaginary part, are used in various areas of mathematics and can be represented in rectangular, polar, or exponential form.
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Why when we consider the inclined angle between two vectors , we must consider the angle anti-clockwisely? What is the reason behind?
 
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No reason except "convention". We had to choose one way or the other- someone flipped a coin!
 

1. What is a vector?

A vector is a mathematical object that represents both magnitude (size) and direction. It is often depicted as an arrow, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction. Vectors are commonly used in physics, engineering, and mathematics to represent quantities such as velocity, force, and displacement.

2. How are vectors represented?

Vectors can be represented in several ways, including using coordinates, components, or unit vectors. In coordinate form, a vector is represented by an ordered list of numbers (e.g. (3,5)). In component form, a vector is represented by its horizontal and vertical components (e.g. (3,5) = 3i + 5j). Unit vectors represent the direction of a vector and are commonly denoted by the symbols i, j, and k for the x, y, and z directions respectively.

3. What operations can be performed on vectors?

Vectors can be added, subtracted, multiplied by a scalar (a single number), and multiplied by another vector using different rules depending on the representation used. These operations are used to find the resultant vector, calculate the magnitude and direction of a vector, and solve problems involving vectors in physics and engineering.

4. What are complex numbers?

Complex numbers are numbers that contain both a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part (b is a real number and i is the imaginary unit, √-1). Complex numbers are used in many areas of mathematics, including algebra, calculus, and geometry, and have applications in physics, engineering, and finance.

5. How are complex numbers represented?

Complex numbers can be represented in several forms, including rectangular form, polar form, and exponential form. In rectangular form, a complex number is written as a + bi, as mentioned before. In polar form, a complex number is written as r(cosθ + isinθ), where r is the magnitude and θ is the angle (in radians) from the positive real axis to the complex number. In exponential form, a complex number is written as re^(iθ), where r and θ have the same meaning as in polar form and e is the base of the natural logarithm.

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