https://file.aiquickdraw.com/mathbot/file/1725935605278bhlby28s.jpg

To solve the problem, let's summarize the given information and determine the first displacement vector \(\vec{A}\):

1. **Displacement \(\vec{B}\)**:
   - Magnitude: \(6.37 \, \text{cm}\)
   - Direction: \(62.9^\circ\) measured from the \(+x\)-axis toward the \(-y\)-axis.

2. **Resultant \(\vec{C} = \vec{A} + \vec{B}\)**:
   - Magnitude: \(6.37 \, \text{cm}\)
   - Direction: \(22.5^\circ\) measured from the \(+x\)-axis toward the \(+y\)-axis.

### Step 1: Determine Components of \(\vec{B}\)
\[
\vec{B}_x = 6.37 \cos(62.9^\circ) \quad \text{and} \quad \vec{B}_y = 6.37 \sin(62.9^\circ)
\]

### Step 2: Determine Components of \(\vec{C}\)
\[
\vec{C}_x = 6.37 \cos(22.5^\circ) \quad \text{and} \quad \vec{C}_y = 6.37 \sin(22.5^\circ)
\]

### Step 3: Find Components of \(\vec{A}\) Using \(\vec{C} = \vec{A} + \vec{B}\)
\[
\vec{A}_x = \vec{C}_x - \vec{B}_x \quad \text{and} \quad \vec{A}_y = \vec{C}_y - \vec{B}_y
\]

### Step 4: Magnitude and Direction of \(\vec{A}\)
\[
|\vec{A}| = \sqrt{\vec{A}_x^2 + \vec{A}_y^2} \quad \text{and} \quad \theta_A = \tan^{-1}\left(\frac{\vec{A}_y}{\vec{A}_x}\right)
\]

Let's compute these values.Here are the computed values:

1. **Components of \(\vec{B}\)**:
   - \(\vec{B}_x = 2.90 \, \text{cm}\)
   - \(\vec{B}_y = 5.67 \, \text{cm}\)

2. **Components of \(\vec{C}\)**:
   - \(\vec{C}_x = 5.89 \, \text{cm}\)
   - \(\vec{C}_y = 2.44 \, \text{cm}\)

3. **Components of \(\vec{A}\)**:
   - \(\vec{A}_x = 2.98 \, \text{cm}\)
   - \(\vec{A}_y = -3.23 \, \text{cm}\)

4. **Magnitude and Direction of \(\vec{A}\)**:
   - Magnitude: \(4.40 \, \text{cm}\)
   - Direction: \(-47.3^\circ\) (measured clockwise from the \(+x\)-axis).

Would you like further details or have any questions?

### Related Questions:
1. How do you find the resultant vector of two vectors?
2. What is the significance of vector components in physics problems?
3. How can direction angles be interpreted when measured from different axes?
4. How do you convert between polar and rectangular coordinates for vectors?
5. What are some practical applications of vector addition in engineering?

**Tip:** Always double-check the angle measurements for vectors to ensure the correct quadrant interpretation.

You are to program a robotic arm on an assembly line to move in the xy-plane. Its first displacement is A⃗; its second displacement is B⃗, of magnitude 6.37 cm and direction 62.9° measured in the sense from the +x-axis toward the -y-axis. The resultant C⃗ = A⃗ + B⃗ of the two displacements should also have a magnitude of 6.37 cm, but a direction 22.5° measured in the sense from the +x-axis toward the +y-axis.

Learn how to solve a vector addition problem involving displacements of a robotic arm in the xy-plane. The problem includes two displacements B⃗ and C⃗ with given magnitudes and directions. Find the first displacement A⃗ using trigonometric functions and vector component analysis. Suitable for Grades 10-12.

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