The image shows three vectors, $\vec{a}$, $\vec{b}$, and $\vec{c}$, along with their magnitudes and angles:

• $\vec{a}$: Magnitude = $20$, at an angle of $40^\circ$ from the horizontal.
• $\vec{b}$: Magnitude = $20$, at an angle of $35^\circ$ from the horizontal (in the positive x-direction).
• $\vec{c}$: Magnitude = $10$, directed downward (negative y-direction).

The image also contains these questions:

1. $\vec{a} + \vec{b} + \vec{c} = ?$
2. $\vec{a} - \vec{b} - \vec{c} = ?$
3. $|\vec{a} + \vec{b}| = ?$
4. $|\vec{a} + \vec{b} + \vec{c}| = ?$
5. $|\vec{a} + \vec{c}| = ?$

Approach:

1. Decompose each vector into its x and y components: $\vec{a} = (a_x, a_y), \quad \vec{b} = (b_x, b_y), \quad \vec{c} = (c_x, c_y)$
2. Perform vector addition or subtraction.
3. Calculate the resultant vector magnitude using: $|\vec{R}| = \sqrt{R_x^2 + R_y^2}$

I will calculate each step. Let's proceed. math

Vector magnitudes and angles (angles are in degrees)

magnitude_a = 20 angle_a = 40 # degrees

magnitude_b = 20 angle_b = 35 # degrees

magnitude_c = 10 angle_c = 270 # downward, equivalent to -90 degrees

Decompose vectors into x and y components

a_x = magnitude_a * math.cos(math.radians(angle_a)) a_y = magnitude_a * math.sin(math.radians(angle_a))

b_x = magnitude_b * math.cos(math.radians(angle_b)) b_y = magnitude_b * math.sin(math.radians(angle_b))

c_x = magnitude_c * math.cos(math.radians(angle_c)) # cos(270°) = 0 c_y = magnitude_c * math.sin(math.radians(angle_c)) # sin(270°) = -1

Perform operations: vector addition and subtraction

1. a + b + c

R1_x = a_x + b_x + c_x R1_y = a_y + b_y + c_y R1_magnitude = math.sqrt(R1_x2 + R1_y2)

2. a - b - c

R2_x = a_x - b_x - c_x R2_y = a_y - b_y - c_y R2_magnitude = math.sqrt(R2_x2 + R2_y2)

3. |a + b|

R3_x = a_x + b_x R3_y = a_y + b_y R3_magnitude = math.sqrt(R3_x2 + R3_y2)

4. |a + b + c|

R4_magnitude = R1_magnitude # Already calculated in R1

5. |a + c|

R5_x = a_x + c_x R5_y = a_y + c_y R5_magnitude = math.sqrt(R5_x2 + R5_y2)

Results

R1_x, R1_y, R1_magnitude, R2_x, R2_y, R2_magnitude, R3_magnitude, R4_magnitude, R5_magnitude### Results:

1. $\vec{a} + \vec{b} + \vec{c} = (31.70, 14.33)$
   Magnitude: $34.79$

2. $\vec{a} - \vec{b} - \vec{c} = (-1.06, 11.38)$
   Magnitude: $11.43$

3. Magnitude of $\vec{a} + \vec{b}$: $39.96$

4. Magnitude of $\vec{a} + \vec{b} + \vec{c}$: $34.79$ (same as in 1)

5. Magnitude of $\vec{a} + \vec{c}$: $15.58$

Would you like a detailed step-by-step explanation of how each value was calculated?

Follow-up Questions:

1. How do we calculate the direction of a resultant vector?
2. Can you derive the angle between two vectors using their components?
3. What happens if $\vec{c}$ is reversed? How do results change?
4. How can we determine if three vectors are linearly dependent?
5. Can we generalize this process for vectors in 3D space?

Tip:

Always break a vector into components to simplify operations like addition, subtraction, or magnitude calculation.