- Draw and label applicable models with units 2. Vector $$ c $$ has a magnitude of 6 and makes an angle of $$ 15^{\circ} $$ with the horizontal, Vector $$ d $$ has a magnitude of 4 and makes an angle of $$ 66^{\circ} $$ with the horizontal. What is the magnitude of $$ c+ $$ d? Round your answer to the nearest hundredth of a unit. *Non-negotiables*: - Combine the horizontal components of each vector, combine vertical components of each vector - Find the magnitude of the resultant vector (the hypotenuse of the newly created right triangle) by using the Pythagorean Theorem - Use calculator and round appropriately

Question

UpStudy AI · Accepted Answer

**Step 1. Draw and Label the Vectors**

- **Vector $$ c $$:**
  - Magnitude: $$ 6 $$ units
  - Angle: $$ 15^{\circ} $$ above the horizontal
  - Horizontal component: $$ c_x=6\cos(15^{\circ}) $$
  - Vertical component: $$ c_y=6\sin(15^{\circ}) $$

- **Vector $$ d $$:**
  - Magnitude: $$ 4 $$ units
  - Angle: $$ 66^{\circ} $$ above the horizontal
  - Horizontal component: $$ d_x=4\cos(66^{\circ}) $$
  - Vertical component: $$ d_y=4\sin(66^{\circ}) $$

*On a diagram, plot a horizontal axis (x-axis) and a vertical axis (y-axis). Draw vector $$ c $$ starting from the origin, making an angle of $$15^{\circ}$$ with the horizontal. Similarly, draw vector $$ d $$ beginning from the origin with an angle of $$66^{\circ}$$ with the horizontal. To add the vectors, place the tail of vector $$ d $$ at the head of vector $$ c $$, or equivalently, compute the resultant by adding the components.*

---

**Step 2. Combine the Horizontal and Vertical Components**

- **Horizontal components:**

$$
  R_x = c_x + d_x = 6\cos(15^{\circ}) + 4\cos(66^{\circ})
  $$

- **Vertical components:**

$$
  R_y = c_y + d_y = 6\sin(15^{\circ}) + 4\sin(66^{\circ})
  $$

---

**Step 3. Compute the Components Using a Calculator**

- Compute $$ 6\cos(15^{\circ}) $$:

$$
  6 \times \cos(15^{\circ}) \approx 6 \times 0.9659 \approx 5.80
  $$

- Compute $$ 4\cos(66^{\circ}) $$:

$$
  4 \times \cos(66^{\circ}) \approx 4 \times 0.4067 \approx 1.63
  $$

- **Sum of horizontal components:**

$$
  R_x \approx 5.80 + 1.63 = 7.43
  $$

- Compute $$ 6\sin(15^{\circ}) $$:

$$
  6 \times \sin(15^{\circ}) \approx 6 \times 0.2588 \approx 1.55
  $$

- Compute $$ 4\sin(66^{\circ}) $$:

$$
  4 \times \sin(66^{\circ}) \approx 4 \times 0.9135 \approx 3.65
  $$

- **Sum of vertical components:**

$$
  R_y \approx 1.55 + 3.65 = 5.20
  $$

---

**Step 4. Find the Magnitude of the Resultant Vector**

Using the Pythagorean theorem, the magnitude $$ R $$ of the resultant vector is given by:

$$
R = \sqrt{R_x^2 + R_y^2}
$$

Substitute the computed values:

$$
R = \sqrt{(7.43)^2 + (5.20)^2} \approx \sqrt{55.23 + 27.04} \approx \sqrt{82.27}
$$

$$
R \approx 9.07 \text{ units}
$$

---

**Final Answer**

The magnitude of $$ c+d $$ is approximately $$ 9.07 $$ units.
The magnitude of $$ c + d $$ is approximately 9.07 units.

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