Advertisements
Advertisements
प्रश्न
Find the modulus and amplitude of the following complex number.
−4 − 4i
Advertisements
उत्तर
Let z = −4 − 4i.
Here, a = −4, b = −4 i.e., a < 0, b < 0
∴ |z| = `sqrt(a^2 + b^2)`
= `sqrt((-4)^2 + (-4)^2`
= `sqrt(16 + 16)`
= `sqrt(32)`
= `4sqrt(2)`
Here, (−4, −4) lies in 3rd quadrant.
The amplitude (or argument) of a complex number z = a + bi is the angle θ formed with the positive real axis in the complex plane, and it is given by:
∴ θ = `tan^-1(b/a)`
= `tan^-1((-4)/(-4))`
= `tan^-1(1)`
= `tan^-1 (tan pi/4)`
θ = `pi/4`
However, since the complex number −4−4i lies in the third quadrant (both real and imaginary parts are negative), the angle needs to be adjusted. The angle in the third quadrant is:
θ = `pi + pi/4`
θ = `(5pi)/4`
Hence, modulus = `4sqrt2` and
Amplitude (θ) = `(5pi)/4`
APPEARS IN
संबंधित प्रश्न
Find the modulus and amplitude of the following complex numbers.
7 − 5i
Find the modulus and amplitude of the following complex numbers.
−3(1 − i)
Find the modulus and amplitude of the following complex numbers.
`sqrt(3) - "i"`
Find the modulus and amplitude of the following complex numbers.
3
Find the modulus and amplitude of the following complex numbers.
1 + i
Find the modulus and amplitude of the following complex numbers.
`1 + "i"sqrt(3)`
If z = 3 + 5i then represent the `"z", bar("z"), - "z", bar(-"z")` in Argand's diagram
Express the following complex numbers in polar form and exponential form:
`-1 + sqrt(3)"i"`
Express the following complex numbers in polar form and exponential form:
−1
Express the following complex numbers in polar form and exponential form:
`(1 + 2"i")/(1 - 3"i")`
Express the following numbers in the form x + iy:
`sqrt(2)(cos (7pi)/4 + "i" sin (7pi)/4)`
Express the following numbers in the form x + iy:
`"e"^((5pi)/6"i")`
For z = 2 + 3i verify the following:
`bar((bar"z"))` = z
For z = 2 + 3i verify the following:
`"z"bar("z")` = |z|2
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar("z"_1 + "z"_2) = bar("z"_1) + bar("z"_2)`
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar(("z"_1/"z"_2))=bar("z"_1)/bar("z"_2)`
Select the correct answer from the given alternatives:
If arg(z) = θ, then arg `bar(("z"))` =
Select the correct answer from the given alternatives:
If `-1 + sqrt(3)"i"` = reiθ , then θ = .................
Select the correct answer from the given alternatives:
If z = x + iy and |z − zi| = 1 then
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
8 + 15i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
6 − i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`(-1 - "i")/sqrt(2)`
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`1/sqrt(2) + 1/sqrt(2)"i"`
Answer the following:
Convert the complex numbers in polar form and also in exponential form.
z = `-6 + sqrt(2)"i"`
The polar coordinates of the point whose cartesian coordinates are (−2, −2), are given by ____________.
The modulus of z = `sqrt7` + 3i is ______
The modulus and amplitude of 4 + 3i are ______
If x + iy = `5/(3 + costheta + isintheta)`, then x2 + y2 is equal to ______
If z = `π/4(1 + i)^4((1 - sqrt(π)i)/(sqrt(π) + i) + (sqrt(π) - i)/(1 + sqrt(π)i))`, then `(|z|/("amp"^((z))))` is equals to ______.
If A, B, C are three points in argand plane representing the complex numbers z1, z2 and z3 such that, z1 = `(λz_2 + z_3)/(λ + 1)`, where λ ∈ R, then find the distance of point A from the line joining points B and C.
