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11 January, 19:11

Calculate z. w for the given complex numbers z and w. Verify that the length, resp. angle, of z. w is the product of lengths, resp. sum of angles, of z and w. (a) z=1-j; w=2j (b) z=3+j; w=-1-2j

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  1. 11 January, 19:30
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    a. Given that z=1-j and w=2j

    Then, z. w = 1*0+-1*2 = 0-2 = -2

    Then, z. w=-2. Answer

    Also, z. w = |z||w|Cosθ

    |z| = sqrt (1^2 + (-1) ^2) = √2

    |w| = sqrt (0^2+2^2) = √4=2

    Therefore, Cosθ = z. w/|z||w|

    Cosθ=-2 / (2*√2) = 1/√2

    θ=arcCos (1/√2) = 45

    For z=1-j

    The angle is arctan (y/x)

    Arctan (-1/1) = - 45

    For w=2j

    The angle is arctan (y/x)

    Arctan (2/0) = 90.

    Sum of angle is - 45+90 = 45

    Therefore, the sum of the individual angles z and w is equal to the angle of z. w

    |z||w| = 2 * √2=2.83

    This product of the individual length of z and w is not equal to the length z. w, this is because the angle between them is 45 assuming the angle is 0 then it will be equal.

    Or this can also happen when the two lines are parallel.

    b. Given that z=3+j and w=-1-2j

    Then, z. w = 3*-1+1*-2 = - 3-2 = -5

    Then, z. w=-5. Answer

    Also, z. w = |z||w|Cosθ

    |z| = sqrt (3^2+1^2) = √10

    |w| = sqrt ((-1) ^2 + (-2) ^2) = √5=

    Therefore, Cosθ = z. w/|z||w|

    Cosθ=-5/√50

    θ=arcCos (-5/√50) = 135 or - 225

    For z=3+j

    The angle is arctan (y/x)

    Arctan (1/3) = 18.43

    For w=-1-2j

    The angle is arctan (y/x)

    Arctan (-2/-1) = - 243.43

    Sum of angle = - 243.43+18.43 = - 225

    Therefore, the sum of the individual angles z and w is equal to the angle of z. w

    |z||w| = √5 * √10=7.07

    This product of the individual length of z and w is not equal to the length z. w, this is because the angle between them is 135 assuming the angle is 0 then it will be equal.
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