Let
𝑢
=
1
−
3
𝑖
and
𝑣
=
5
+
𝑖
.
Which of the following is the Cartesian form of the complex number
𝑢
+
𝑢
𝑣
?

Question

Let
𝑢
=
1
−
3
𝑖
 and 
𝑣
=
5
+
𝑖
.
Which of the following is the Cartesian form of the complex number 
𝑢
+
𝑢
𝑣
?

BlackTom AI · Accepted Answer

To determine the Cartesian form of the complex number $ u + uv $, where $ u = 1 - 3i $ and $ v = 5 + i $, we first need to compute $ uv $ and then add $ u $ to the result.

First, calculate $ uv $:
$$
uv = (1 - 3i)(5 + i) = 1 \cdot 5 + 1 \cdot i - 3i \cdot 5 - 3i \cdot i = 5 + i - 15i - 3i^2.
$$
Recall that $ i^2 = -1 $, so $ -3i^2 = 3 $. Thus, the expression becomes:
$$
uv = 5 + i - 15i + 3 = 8 - 14i.
$$

Now, add $ u $ to $ uv $:
$$
u + uv = (1 - 3i) + (8 - 14i) = 1 + 8 - 3i - 14i = 9 - 17i.
$$

Therefore, the Cartesian form of $ u + uv $ is $ 9 - 17i $.

Comparatively, examining the other options:
- $ 9 - 23i $ suggests an incorrect computation of the imaginary part, likely due to an error in combining the imaginary components.
- $ 11 - 15i $ indicates an error in both the real and imaginary parts, possibly from incorrect multiplication or addition.
- $ 9 + 14i $ shows a sign error in the imaginary part, which should be negative due to the subtraction of imaginary components.

Thus, the correct option is $ 9 - 17i $, as it accurately reflects the operations performed on the complex numbers.

Let 𝑢 = 1 − 3 𝑖 and 𝑣 = 5 + 𝑖 . Which of the following is the Cartesian form of the complex number 𝑢 + 𝑢 𝑣 ?单项选择题

类似问题

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