Find the principal argument of z = -1 − i.

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Multiple Choice

Find the principal argument of z = -1 − i.

The principal argument is the angle the vector from the origin to the point (-1, -1) makes with the positive real axis, chosen in the range (-π, π]. Since both coordinates are negative, the point lies in the third quadrant. The reference angle is arctan(|-1|/|-1|) = arctan(1) = π/4. In the third quadrant the standard angle is π + π/4 = 5π/4, but the principal value must lie between -π and π, so subtract 2π to get -3π/4. This places the angle at -135 degrees, matching the direction of (-1, -1). The other angles correspond to different quadrants (first, fourth, and second), so they don’t match the vector’s direction.

The principal argument is the angle the vector from the origin to the point (-1, -1) makes with the positive real axis, chosen in the range (-π, π]. Since both coordinates are negative, the point lies in the third quadrant. The reference angle is arctan(|-1|/|-1|) = arctan(1) = π/4. In the third quadrant the standard angle is π + π/4 = 5π/4, but the principal value must lie between -π and π, so subtract 2π to get -3π/4. This places the angle at -135 degrees, matching the direction of (-1, -1). The other angles correspond to different quadrants (first, fourth, and second), so they don’t match the vector’s direction.

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