Let's **analyze each option** to check if the vector/scalar notation is used correctly. **Vectors** typically use arrows (→), boldface, or unit vectors (like \(\hat{\imath}\), \(\hat{\mathbf{e}}_r\), \(\vec{v}\)), and **scalars** do **not**. --- ### 1. \( \vec{v} - 5 \text{ m/s} \) - **Incorrect**: You can't subtract a vector (\(\vec{v}\)) and a scalar (5). The units match, but vector and scalar addition/subtraction must be consistent. --- ### 2. \( a_A = a_B + a_{A/B} \) (non-rectilinear motion) - **Correct**: All terms are accelerations (vectors); subscripts just indicate different frames or components. This is correct vector notation. --- ### 3. \( v = 45\hat{e}_r + 72\hat{e}_\theta \) - **Correct**: Both terms are multiplied by unit vectors (\(\hat{e}_r\), \(\hat{e}_\theta\)), so this is a proper vector sum. --- ### 4. \( a = 57e_n \) - **Correct**: \(e_n\) is commonly used as a unit vector in the normal direction, so this is appropriate notation for a vector. --- ### 5. \( |\hat{v}| = 26 \text{ m/s} \) - **Incorrect**: The magnitude of a vector should be written as \(|\vec{v}|\) or just \(v\), not \(|\hat{v}|\), because \(\hat{v}\) is a unit vector (its magnitude is always 1). --- ### 6. \( a = 5\,\mathbf{i} + 7\,\mathbf{j} \) - **Correct**: This is proper vector notation in Cartesian coordinates using unit vectors. --- ### 7. \( a_t = 2.6 \text{ m/s}^2 \) - **Correct**: \(a_t\) typically refers to the scalar tangential component of acceleration, so this is correct. --- ### 8. \( \vec{a}_t = 2.6 \hat{e}_t \text{ m/s}^2 \) - **Correct**: This is a vector, with the correct notation using the unit vector in the tangential direction. --- ### 9. \( v = 5 e_\theta \) - **Correct**: \(e_\theta\) is a unit vector, so this is a vector notation for velocity in polar coordinates. --- ## **Final Answers:** ### **Correct Notation:** - [x] 2) \( a_A = a_B + a_{A/B} \) - [x] 3) \( v = 45\hat{e}_r + 72\hat{e}_\theta \) - [x] 4) \( a = 57e_n \) - [x] 6) \( a = 5\,\mathbf{i} + 7\,\mathbf{j} \) - [x] 7) \( a_t = 2.6\,\text{m/s}^2 \) - [x] 8) \( \vec{a}_t = 2.6\,\hat{e}_t\,\text{m/s}^2 \) - [x] 9) \( v = 5e_\theta \) ### **Incorrect Notation:** - [ ] 1) \( \vec{v} - 5\,\text{m/s} \) - [ ] 5) \( |\hat{v}| = 26\,\text{m/s} \) --- **Summary Table:** | Option | Correct? | Reason | |--------|----------|------------------------------------------------------------------------| | 1 | ✗ | Can't subtract vector and scalar | | 2 | ✓ | All are vectors, correct notation | | 3 | ✓ | Vector sum with unit vectors | | 4 | ✓ | Vector with unit vector | | 5 | ✗ | Magnitude of unit vector is always 1 | | 6 | ✓ | Vector in Cartesian form | | 7 | ✓ | Scalar component, correct | | 8 | ✓ | Vector with unit vector, correct | | 9 | ✓ | Vector in polar coordinates, correct |