Appendix G — Problem Set 5 | DIRECTION: Angles and Projections

Companion to §2.3 — the dot product and projection

A. Recognition

1. Dot product, cross product, or neither?

Choose the most direct operation and justify it.

  1. Find the angle between two bonds.
  2. Find the component of an electric field along a bond.
  3. Find a vector normal to a molecular plane.
  4. Add three displacement vectors.
  5. Test whether two directions are perpendicular.

2. Scalar projection or vector projection?

State which is requested.

  1. “How much of \mathbf F acts along the reaction coordinate?”
  2. “What vector is the part of \mathbf F parallel to the reaction coordinate?”
  3. “What remains after the parallel part is removed?”

B. Angles and projections

3. Bond angle from coordinates

An atom B is at the origin; A and C are at \mathbf r_A=(1,1,0) Å and \mathbf r_C=(-1,2,0) Å.

  1. Write vectors \overrightarrow{BA} and \overrightarrow{BC}.
  2. Calculate \angle ABC using the dot product.
  3. Verify that rescaling either bond length without changing its direction leaves the angle unchanged.

4. Electric field along a bond

An electric field is \mathbf E=(2.0,-1.0,3.0)\times10^5\ \mathrm{V\,m^{-1}}. A bond points along \mathbf b=(2,1,2).

  1. Construct \hat{\mathbf b}.
  2. Calculate the signed scalar component E_\parallel=\mathbf E\cdot\hat{\mathbf b}.
  3. Calculate the vector projection \mathbf E_\parallel.
  4. Calculate the perpendicular remainder and verify orthogonality.

5. Work along a molecular displacement

A constant force \mathbf F=(4,-2,1)\ \mathrm{pN} displaces an atom by \Delta\mathbf r=(0.30,0.10,-0.20)\ \mathrm{nm}.

  1. Calculate W=\mathbf F\cdot\Delta\mathbf r in \mathrm{pN\,nm}.
  2. Convert to joules.
  3. Is the force, on balance, assisting or opposing this displacement?

6. Projection decomposition theorem

For nonzero \mathbf b, define \mathbf a_\parallel=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf b, \qquad \mathbf a_\perp=\mathbf a-\mathbf a_\parallel.

  1. Prove \mathbf a_\perp\cdot\mathbf b=0.
  2. Apply the decomposition to \mathbf a=(3,1,2) and \mathbf b=(1,-1,0).
  3. Verify |\mathbf a|^2=|\mathbf a_\parallel|^2+|\mathbf a_\perp|^2.

C. Molecular geometry

7. Plane normal and orientation

Three atoms lie at A(1,0,0), B(0,1,0), and C(0,0,1) Å.

  1. Form \overrightarrow{AB} and \overrightarrow{AC}.
  2. Compute a normal using \overrightarrow{AB}\times\overrightarrow{AC}.
  3. Normalize it.
  4. Explain why the negative of your normal is equally perpendicular but not identically oriented.

8. Torque on a dipole

A dipole \boldsymbol\mu=(2.0,0,0)\ \mathrm D lies in an electric field \mathbf E=(0,3.0,0)\times10^5\ \mathrm{V\,m^{-1}}.

  1. Determine the direction of \boldsymbol\tau=\boldsymbol\mu\times\mathbf E.
  2. Calculate its magnitude in symbolic units \mathrm{D\,V\,m^{-1}}.
  3. What happens to torque when the dipole becomes parallel or antiparallel to the field?
  4. Which orientation minimizes U=-\boldsymbol\mu\cdot\mathbf E?

9. Tetrahedral check

Four ideal tetrahedral directions may be represented by (1,1,1),\ (1,-1,-1),\ (-1,1,-1),\ (-1,-1,1).

  1. Normalize any two.
  2. Show their dot product is -1/3.
  3. Calculate the angle and compare it with the tetrahedral angle 109.47^\circ.
  4. Explain why every pair gives the same angle.

10. Error diagnosis: inverse cosine

A student finds \mathbf a\cdot\mathbf b=-2, |\mathbf a|=2, and |\mathbf b|=2, then reports \theta=-0.5^\circ.

  1. Identify both conceptual errors.
  2. Calculate the correct angle.
  3. State the allowable range of (\mathbf a\cdot\mathbf b)/(|\mathbf a||\mathbf b|) and why.