a=150g, 20<Degrees. b=377g, 120<degrees. c=225g<260degrees. d=95g, <340degrees. solve magnitude and direction of E.  1. A+B+C+E=0. 

The given problem involves adding vectors to solve for the magnitude and direction of vector \( E \), such that the equation \( A + B + C + D + E = 0 \) holds true. Here's a step-by-step breakdown:

### Step 1: Express Vectors in Component Form
The vectors are given in terms of their magnitudes and directions. We will break them into their \( x \)- and \( y \)-components using trigonometry.

For a vector with magnitude \( M \) and direction \( \theta \), the components are:
- \( M_x = M \cdot \cos(\theta) \)
- \( M_y = M \cdot \sin(\theta) \)

We will convert the angles to radians for computation.

#### 1. Vector A (150g, 20°)
- \( A_x = 150 \cdot \cos(20^\circ) \)
- \( A_y = 150 \cdot \sin(20^\circ) \)

#### 2. Vector B (377g, 120°)
- \( B_x = 377 \cdot \cos(120^\circ) \)
- \( B_y = 377 \cdot \sin(120^\circ) \)

#### 3. Vector C (225g, 260°)
- \( C_x = 225 \cdot \cos(260^\circ) \)
- \( C_y = 225 \cdot \sin(260^\circ) \)

#### 4. Vector D (95g, 340°)
- \( D_x = 95 \cdot \cos(340^\circ) \)
- \( D_y = 95 \cdot \sin(340^\circ) \)

### Step 2: Sum of Components
The vector equation \( A + B + C + D + E = 0 \) implies that the sum of the components of vectors \( A \), \( B \), \( C \), and \( D \) should give the negative of vector \( E \) in both the \( x \)- and \( y \)-directions.

- \( E_x = -(A_x + B_x + C_x + D_x) \)
- \( E_y = -(A_y + B_y + C_y + D_y) \)

### Step 3: Magnitude and Direction of Vector E
Once we have \( E_x \) and \( E_y \), the magnitude and direction of \( E \) are calculated as:
- \( |E| = \sqrt{E_x^2 + E_y^2} \)
- \( \theta_E = \tan^{-1}\left(\frac{E_y}{E_x}\right) \)

Let me now compute these values for you.The magnitude and direction of vector \( E \) are as follows:
- Magnitude: \( 123.75 \, g \)
- Direction: \( -91.23^\circ \) (or 268.77° if you prefer a positive angle)

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do the components of a vector change when the angle is adjusted?
2. How does vector addition work geometrically in two dimensions?
3. How do we interpret negative angles in vector direction?
4. What are real-world applications of vector addition?
5. How do the sine and cosine functions relate to vector components?

**Tip:** Always express vector directions in terms of standard angles (positive counterclockwise from the positive x-axis).

a=150g, 20<Degrees. b=377g, 120<degrees. c=225g<260degrees. d=95g, <340degrees. solve magnitude and direction of E. 1. A+B+C+E=0.

Learn how to solve a vector addition problem involving multiple vectors with different magnitudes and directions. This solution demonstrates how to calculate the components, magnitude, and direction of vector E, using trigonometry and vector decomposition techniques. Suitable for students in Grades 10-12.

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