Compute the magnitude and argument of z = -3 + 3i.

Boost your skills in A Level Further Mathematics Core Pure. Study with flashcards and multiple choice questions, each question is followed by hints and explanations. Get prepared for your exam with confidence!

Multiple Choice

Compute the magnitude and argument of z = -3 + 3i.

Finding the magnitude and argument of a complex number means first measuring its length from the origin and then the angle its vector makes with the positive real axis. For z = -3 + 3i, the magnitude is |z| = sqrt((-3)^2 + 3^2) = sqrt(9 + 9) = sqrt(18) = 3√2. The argument is the angle of the vector (-3, 3) in the complex plane. The reference angle is arctan(|3|/|−3|) = arctan(1) = π/4. Since the point lies in the second quadrant (x negative, y positive), the true angle from the positive real axis is π − π/4 = 3π/4. So Arg z = 3π/4. Thus the magnitude and argument are 3√2 and 3π/4, which matches the correct option. The other possibilities either give a different length or place the angle in a different quadrant, corresponding to different complex numbers.

Finding the magnitude and argument of a complex number means first measuring its length from the origin and then the angle its vector makes with the positive real axis.

For z = -3 + 3i, the magnitude is |z| = sqrt((-3)^2 + 3^2) = sqrt(9 + 9) = sqrt(18) = 3√2.

The argument is the angle of the vector (-3, 3) in the complex plane. The reference angle is arctan(|3|/|−3|) = arctan(1) = π/4. Since the point lies in the second quadrant (x negative, y positive), the true angle from the positive real axis is π − π/4 = 3π/4. So Arg z = 3π/4.

Thus the magnitude and argument are 3√2 and 3π/4, which matches the correct option. The other possibilities either give a different length or place the angle in a different quadrant, corresponding to different complex numbers.

More practice questions