explain complex number and vectors
in Word Problem Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Start with real numbers. A number line is often used to represent all real numbers. It has infinite length and somewhere we can mark zero, dividing positive numbers (on the right of zero) and negative numbers (on the left of zero). The line is continuous so no real number can be left out. A line is 1-dimensional.

A complex number has two components: one real, the other imaginary. A complex number can be represented by a plane, and it's 2-dimensional. So what does imaginary mean? The basis of imaginary numbers is the square root of minus one (sqrt(-1)) and traditionally it is given the symbol i. The square root of any negative number can be expressed using i. So, for example, the square root of minus 4 is 2i, because -4=4*-1 and the square root of 4*-1 is 2sqrt(-1)=2i. A complex number, z, can be written a+ib, where a and b are real numbers. But, more importantly perhaps, they can be represented as a point in 2-dimensional space as the point (a,b) plotted on a graph using the familiar x-y coordinate system. So, just like the number line represented all real numbers like an x axis, so all complex numbers can be represented by a plane, an infinite x-y plane.

Now we come to vectors. There is a commonality between complex numbers and vectors. A straight road with cars , houses, people, etc., on it make the road like a number line. Any position on the road can be related to a fixed point on the road we'll call "home". Objects to the right, or eastward, could be in front of home and those to the left, or westward, behind home. The position of an object is the distance from home. This is a 1-dimensional vector field, where all objects have position. If the objects move their speed will have direction, towards the right or towards the left and we can say that a speed to the right is positive and a speed to the left is negative.

Now we introduce another straight road at right angles to the first road. Now the picture is 2-dimensional. The position of an object is defined by two values: position east or west and position north or south. North can be positive and south negative. This is equivalent to the complex plane, which represents all complex numbers. The 2-dimensional plane represents all 2-dimensional vectors, whether it's position or speed. But the word "velocity" is used instead of "speed", because velocity includes direction, but speed is just a number, or magnitude, of the velocity. A vector has an east-west (EW) component (x value) and a north-south (NS) component (y value) and a vector r=xi+yj, where i is called a unit vector in the NS direction and j in the EW direction, so the point (x,y) fixes the positional vector r. Vectors are usually written in bold type, so you won't confuse i with i. The magnitude of a vector is sqrt(x^2+y^2) so it is represented by the hypotenuse of a right-angled triangle whose other sides are x and y. Pythagoras' theorem is used to work out the value. The magnitude of a vector is sometimes written |r| and is called a "scalar" quantity, so it doesn't have a direction or a sign (positive or negative), because the sign is a property of the direction of the vector. 

A vector is not limited to a 2-dimensional plane. It can have as many dimensions as necessary. An aeroplane's positional vector and velocity would involve another dimension: height. A submarine's positional vector and velocity would involve depth. Height and depth are perpendicular to the EWNS plane and together form 3-dimensional space. The unit vector for height and depth is k, and height would be positive while depth would be negative, and the letter z is used with x and y so that a point in 3-space is (x, y, z). The magnitude |r| is sqrt(x^2+y^2+z^2).

When working with vectors, addition and subtraction requires adding and subtracting the x, y, z components separately. When adding or subtracting complex numbers, the same applies to the x and y, real and complex, components. Multiplication and division are a special topic beyond the scope of this introductory explanation.

 

by Top Rated User (1.1m points)

Related questions

1 answer
asked Jan 31, 2017 in Other Math Topics by anonymous | 733 views
1 answer
1 answer
asked Jul 31, 2014 in Other Math Topics by anonymous | 548 views
1 answer
1 answer
1 answer
0 answers
2 answers
asked Jan 13, 2012 in Calculus Answers by anonymous | 1.5k views
1 answer
asked Jun 12, 2018 in Algebra 2 Answers by Subarna Das Level 1 User (440 points) | 342 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method proving trigonometric identities least common multiple factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: mathhomeworkanswers.org ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than geometry division fractions how to find x intercept differentiation exponents 8th grade math simplifying fractions geometry 10th grade equivalent fractions inverse function area of a triangle elimination method story problems standard deviation integral ratios simplify systems of equations containing three variables width of a rectangle percentages area of a circle circumference of a circle place value solving triangles parallel lines mathematical proofs solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems quadratic functions number of sides of a polygon length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets set builder notation please help me to answer this step by step significant figures simplifying radicals arithmetic sequences median age problem trigonometry graphing derivatives number patterns adding fractions radicals midpoint of a line roots of polynomials product of two consecutive numbers limits decimals compound interest please help pre-algebra problems divisibility rules graphing functions subtracting fractions angles numbers discrete mathematics volume of a cylinder simultaneous equations integration probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 matrices logarithms how to complete the square mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,441 questions
99,039 answers
2,422 comments
16,939 users