Resultant of Two Vectors in the Plane

We are given two vectors in polar form:

• Vector A: magnitude $|\vec A| = 5$ at angle $30^\circ$
• Vector B: magnitude $|\vec B| = 8$ at angle $120^\circ$

We want their resultant vector:

$$\vec R = \vec A + \vec B.$$

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1. Convert each vector to components

For a vector with magnitude $r$ and angle $\theta$ (in degrees, measured from +x axis):

$$x = r\cos\theta, \quad y = r\sin\theta.$$

Vector A

$$A_x = 5 \cos 30^\circ, \quad A_y = 5 \sin 30^\circ.$$

Using exact values:

• $\cos 30^\circ = \sqrt{3}/2 \approx 0.866$
• $\sin 30^\circ = 1/2 = 0.5$

So:

$$A_x = 5 \cdot 0.866 \approx 4.33, \quad A_y = 5 \cdot 0.5 = 2.5.$$

Vector B

$$B_x = 8 \cos 120^\circ, \quad B_y = 8 \sin 120^\circ.$$

Using:

• $\cos 120^\circ = -1/2 = -0.5$
• $\sin 120^\circ = \sqrt{3}/2 \approx 0.866$

So:

$$B_x = 8 \cdot (-0.5) = -4, \quad B_y = 8 \cdot 0.866 \approx 6.93.$$

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2. Add the components

The resultant components:

$$R_x = A_x + B_x, \quad R_y = A_y + B_y.$$

Plugging in values:

$$R_x = 4.33 + (-4) \approx 0.33,\\ R_y = 2.5 + 6.93 \approx 9.43.$$

So the resultant vector in component form is approximately:

$$\vec R \approx (0.33,\; 9.43).$$

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3. Magnitude and direction of the resultant

Magnitude:

$$|\vec R| = \sqrt{R_x^2 + R_y^2} \approx \sqrt{0.33^2 + 9.43^2} \approx \sqrt{0.11 + 88.94} \approx \sqrt{89.05} \approx 9.43.$$

Direction (angle from +x axis):

$$\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) \approx \tan^{-1}\left(\frac{9.43}{0.33}\right) \approx 88^\circ.$$

So the resultant is almost vertical, with a small positive x-component:

$$\boxed{|\vec R| \approx 9.4 \text{ units at } 88^\circ.}$$

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4. What the visualization shows

The interactive canvas illustrates:

1. Vector A (cyan): drawn from the origin at angle $\theta_A$ with length proportional to its magnitude.
2. Vector B (pink): also from the origin at angle $\theta_B$.
3. Resultant vector R (yellow): the sum $\vec A + \vec B$, drawn from the origin to the tip of the parallelogram/triangle formed by A and B.
4. A faint dynamic grid to provide coordinate context.

You can adjust:

• The magnitude and angle of each vector and see how:
  • Their components change.
  • The resultant direction and length updates in real time.

Mathematically, the visualization implements:

$$\begin{aligned} A_x &= |A|\cos(\theta_A), & A_y &= |A|\sin(\theta_A),\\ B_x &= |B|\cos(\theta_B), & B_y &= |B|\sin(\theta_B),\\ R_x &= A_x + B_x, & R_y &= A_y + B_y. \end{aligned}$$

Then it draws the three vectors on a Cartesian plane (transformed to canvas coordinates) with a dark-slate background and neon colors, and slight animation to keep the scene visually alive.