Let's solve this step-by-step. The problem involves multiplying two complex numbers, $z$ and $\sqrt{3} + i$. We need to find which point on the graph corresponds to their product.

Step 1: Recall the multiplication of complex numbers

If you multiply two complex numbers in polar form:

• Magnitude: Multiply their magnitudes.
• Argument (Angle): Add their arguments.

Given the two complex numbers:

1. $z = r \, e^{i\theta}$, with magnitude $r$ and argument $\theta$.
2. $\sqrt{3} + i$, which we rewrite in polar form.

Step 2: Convert $\sqrt{3} + i$ to polar form

1. Magnitude $| \sqrt{3} + i |$: $| \sqrt{3} + i | = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = 2$

2. Argument $\arg(\sqrt{3} + i)$: $\tan^{-1} \left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \, \text{(30 degrees)}$

Thus, $\sqrt{3} + i = 2 \, e^{i\pi/6}$.

Step 3: Product of $z$ and $\sqrt{3} + i$

• Magnitude of product: If $z$ has magnitude $r$, the product will have magnitude $2r$ (since $|z| \cdot | \sqrt{3} + i | = r \cdot 2$).

• Angle of product: If the original angle of $z$ is $\theta$, the new angle will be $\theta + \pi/6$.

Step 4: Locate the product on the graph

1. Check the location of point $z$:

   • $z$ is on the imaginary axis, so its angle $\theta = \pi/2$.

2. New angle after adding $\pi/6$: $\theta + \frac{\pi}{6} = \frac{\pi}{2} + \frac{\pi}{6} = \frac{2\pi}{3} \, \text{(120 degrees)}$ This new angle places the product in the second quadrant.

3. Magnitude of the product:

   • If the magnitude of $z$ is $r$, the new magnitude is $2r$.

Step 5: Identify the correct point

Looking at the points labeled $A, B, C, D$:

• The product will have an angle of $120^\circ$, which corresponds to the second quadrant.
• Among the given points, B is the only one that lies in the second quadrant, aligned with the new angle.

Answer:

The correct answer is:

$\boxed{B}$

Do you have any questions or need further clarification?

5 Related Questions:

1. How do you convert any complex number into polar form?
2. What are the advantages of multiplying complex numbers using polar form?
3. How do arguments behave when adding or subtracting multiple complex numbers?
4. What is the geometric interpretation of multiplying complex numbers?
5. How do angles shift when multiplying with $i$ specifically?

Tip: Whenever you multiply complex numbers, try converting them to polar form if angles and rotations are involved—it makes the calculations much easier.