To verify these vector equations geometrically, we can explain the concepts step by step:

i) $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

Geometric Interpretation:

1. Left-Hand Side (LHS):

   • The dot product $\vec{a} \cdot (\vec{b} + \vec{c})$ measures how much the vector $\vec{a}$ aligns with the sum of $\vec{b} + \vec{c}$.
   • $(\vec{b} + \vec{c})$ represents the diagonal of a parallelogram formed by $\vec{b}$ and $\vec{c}$.

2. Right-Hand Side (RHS):

   • Breaking $\vec{a} \cdot (\vec{b} + \vec{c})$ into $\vec{a} \cdot \vec{b}$ and $\vec{a} \cdot \vec{c}$:
     • $\vec{a} \cdot \vec{b}$: Projection of $\vec{b}$ onto $\vec{a}$.
     • $\vec{a} \cdot \vec{c}$: Projection of $\vec{c}$ onto $\vec{a}$.

Verification:

• Geometrically, the projections add up because the sum $(\vec{b} + \vec{c})$ distributes the effect of the projections over $\vec{a}$.

ii) $\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}$

Geometric Interpretation:

1. LHS:

   • The expression $\vec{a} + (\vec{b} + \vec{c})$ represents adding $\vec{a}$ to the vector sum $\vec{b} + \vec{c}$.
   • Geometrically, this is moving along $\vec{a}$, then along $\vec{b} + \vec{c}$ in sequence.

2. RHS:

   • The expression $(\vec{a} + \vec{b}) + \vec{c}$ means first adding $\vec{a}$ and $\vec{b}$ to get their resultant, then adding $\vec{c}$.
   • Geometrically, this also leads to the same endpoint since vector addition is associative.

Verification:

• Both sides represent reaching the same point geometrically when vectors are added tip-to-tail. The order of addition doesn't change the result.

Visualization Tips:

• To verify these properties geometrically:
  1. Draw vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.
  2. For the first equation, draw the parallelogram formed by $\vec{b}$ and $\vec{c}$.
  3. For the second equation, visualize the tip-to-tail addition of vectors.

Would you like a step-by-step diagram or need further clarification?

Related Questions:

1. How can the distributive property of dot products be proven algebraically?
2. How does the triangle law of vector addition relate to the second equation?
3. What are real-life applications of verifying such vector properties?
4. How do vector operations change in 3D space compared to 2D?
5. Can the associative property of vector addition be applied to cross products as well?

Tip:

Always visualize vector equations