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3 years ago

Explain the geometric efect of z if you multiply z by ????????, where ???????? = 1 + i

Mathematics

1 answer:

BartSMP [9]3 years ago

8 0

Answer:

You will obtain a complex number rotated by an angle of 45 degrees (counterclockwise) with a modulus scaled by √2

Step-by-step explanation:

In order to see the effect of multiplying z by 1 + i, you can use the representation of complex numbers in the Polar Form. This representation gives you the angle formed by the complex number and the real axis and the distance from the origin to the point.

Let z=a+ib represent a complex number. The Polar Form is:

z = |z| (Cosα + iSinα)

Where |z| is the modulus of the complex number and α is the angle formed with the real axis.

|z| = √a²+b²

α= arctan (b/a)

The multiplication in Polar Form is:

Let Z0 and Z1 represent two complex numbers

Z0= |Z0| (Cosα + iSinα)

z1= |z1|(Cosβ + iSinβ)

The multiplication is:

Z0.Z1 = |Z0||Z1| [Cos(α+β) + i Sin(α+β)]

Notice that when you multiply complex numbers, you are adding angles and multiplying modulus. The addition of angles can be seen as a rotation of the complex number on the plane and the multiplication of modulus can be seen as changing the modulus of the complex number.

The given number 1+i in the Polar Form is:

z = |z| (Cosα + i Sinα)

|z| = √1²+1² = √2

α = arctan(1/1) = 45°

Therefore, you will obtain a complex number rotated by an angle of 45 degrees with a modulus scaled by √2

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Step-by-step explanation:

Let's name the smaller number x.

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3 years ago

In the expansion (ax+by)^7, the coefficients of the first two terms are 128 and -224, respectively. Find the values of a and b

madam [21]

Answer:

a = 2, b = 3.5

Step-by-step explanation:

Expanding $(ax+by)^7$ using Binomial expansion, we have that:

$(ax+by)^7$ =

$(ax)^7(by)^0 + (ax)^6(by)^1 + (ax)^5(by)^2 + (ax)^4(by)^3 + (ax)^3(by)^4 + (ax)^2(by)^5 + (ax)^1(by)^6 + (ax)^0(by)^7$

$= (a)^7(x)^7+ (a)^6(x)^6(b)(y) + (a)^5(x)^5(b)^2(y)^2 + (a)^4(x)^4(b)^3(y)^3 + (a)^3(x)^3(b)^4(y)^4 + (a)^2(x)^2(b)^5(y)^5 + (a)(x)(b)^6(y)^6 + (b)^7(y)^7\\\\\\= (a)^7(x)^7+ (a)^6(b)(x)^6(y) + (a)^5(b)^2(x)^5(y)^2 + (a)^4(b)^3(x)^4(y)^3 + (a)^3(b)^4(x)^3(y)^4 + (a)^2(b)^5(x)^2(y)^5 + (a)(b)^6(x)(y)^6 + (b)^7(y)^7$

We have that the coefficients of the first two terms are 128 and -224.

For the first term:

=> $a^7 = 128$

=> $a = \sqrt[7]{128}\\ \\\\a = 2$

For the second term:

$a^6b = -224$

$b = \frac{-224}{a^6}$

$b = \frac{-224}{2^6} \\\\\\b = \frac{-224}{64} \\\\\\b = 3.5$

Therefore, a = 2, b = 3.5

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