Question

Consider the numbers z = startroot 2 endroot (cos 45o i sin 45o )and w = 2(cos 90o i sin 90o). Fill in the _________s to complete the statements. in rectangular form, z − w = 1 − 1 i in polar form z – w = startroot 2 endroot (cos 135 o i sin 135 o)

Answer 1

To find the rectangular form of z - w, subtract the real and imaginary parts of the two complex numbers separately. In this case, z - w = 1 - i. In polar form, find the magnitude and argument of z - w using the formula |z - w| = |z| * |w| and arg(z - w) = arg(z) - arg(w). In this case, z - w is √2(cos -45° + i sin -45°).

Step-by-step explanation:

To find the rectangular form of z - w, we subtract the real parts and imaginary parts of the two complex numbers separately. The real part of z - w is equal to the real part of z minus the real part of w, and the imaginary part is equal to the imaginary part of z minus the imaginary part of w. In this case, z = √2(cos 45° + i sin 45°) and w = 2(cos 90° + i sin 90°).

Real part of z - w = √2(cos 45°) - 2(cos 90°) = 1

Imaginary part of z - w = √2(sin 45°) - 2(sin 90°) = -1i

Therefore, the rectangular form of z - w is 1 - 1i.

In polar form, we can find the magnitude and argument of z - w by using the formula |z - w| = |z| * |w| and arg(z - w) = arg(z) - arg(w). In this case, |z - w| = √2 and arg(z - w) = 45° - 90° = -45°.

Therefore, in polar form, z - w = √2(cos -45° + i sin -45°).