Advertisements
Advertisements
प्रश्न
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
6 − i
Advertisements
उत्तर
Let z = 6 − i
∴ a = 6, b = − 1, a > 0, b < 0
∴ |z| = r
= `sqrt("a"^2 + "b"^2)`
= `sqrt(6^2 + (-1)^2)`
= `sqrt(36 + 1)`
= `sqrt(37)`
Here (6, −1) lies in 4th quadrant
∴ amp (z) = θ = `2pi + tan^-1 ("b"/"a")`
= `2pi + tan^-1 ((-1)/6)`
∴ the polar form of z = r(cos θ + i sin θ)
= `sqrt(37)(cos theta + "i" sin theta)`
where θ = `2pi + tan^-1 (-1/6)`
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.
−8 + 15i
Find the modulus and amplitude of the following complex number.
−4 − 4i
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"sqrt(3)`
Find the modulus and amplitude of the following complex numbers.
(1 + 2i)2 (1 − i)
Find real values of θ for which `((4 + 3"i" sintheta)/(1 - 2"i" sin theta))` is purely real.
Express the following complex numbers in polar form and exponential form:
`1/(1 + "i")`
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(3)(cos pi/6 + "i" sin pi/6)`
Express the following numbers in the form x + iy:
`7(cos(-(5pi)/6) + "i" sin (- (5pi)/6))`
Express the following numbers in the form x + iy:
`"e"^(pi/3"i")`
Express the following numbers in the form x + iy:
`"e"^((-4pi)/3"i")`
Convert the complex number z = `("i" - 1)/(cos pi/3 + "i" sin pi/3)` in the polar form
For z = 2 + 3i verify the following:
`bar((bar"z"))` = z
For z = 2 + 3i verify the following:
`("z" + bar"z")` is real
For z = 2 + 3i verify the following:
`"z" - bar"z"` = 6i
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)`
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"))` =
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.
`(1 + sqrt(3)"i")/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 = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`
Answer the following:
Convert the complex numbers in polar form and also in exponential form.
z = `-6 + sqrt(2)"i"`
Convert the complex numbers in polar form and also in exponential form.
`(-3)/2 + (3sqrt(3))/2"i"`
For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5, the minimum value of |z1 - z2| is ______.
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.
