Consider the following statements :

1. Dot product over vector addition is distributive

2. Cross product over vector addition is distributive

3. Cross product of vectors is associative

Which of the above statements is/are correct ?

Concept:

One algebraic property of real numbers is the distributive law. The distributive law for the real numbers says: "For all real numbers x, y, and z, \(x.( y+ z)=x. y+ x.z\)

The vector dot product is distributive over addition. In general: \(\vec a.( \vec b+ \vec c)=\vec a. \vec b+ \vec a.\vec c\)  

The vector cross product is distributive over addition. In general: \(\vec a × (\vec b + \vec c) = \vec a × \vec b + \vec a × \vec c\)

Associative property: (p × q) × r = p × (q × r) (where p, q, and r are any three natural/whole numbers)

Calculation:

Let

\(\displaystyle \vec a = a_x \overline i+a_y \overline j+a_z \overline k\\\vec b = b_x \overline i+b_y \overline j+b_z \overline k\\\vec c = c_x \overline i+c_y \overline j+c_z \overline k\)

Statement I: Dot product over vector addition is distributive

We have to prove \(\vec a.( \vec b+ \vec c)=\vec a. \vec b+ \vec a.\vec c\)

• \(\displaystyle \vec a.( \vec b+ \vec c)=(a_x ̂ i+a_y ̂ j+a_z ̂ k).[(b_x ̂ i+b_y ̂ j+b_z ̂ k)+(c_x ̂ i+c_y ̂ j+c_z ̂ k)] \)
• \(\displaystyle \vec a.( \vec b+ \vec c)=(a_x ̂ i+a_y ̂ j+a_z ̂ k).[(b_x+c_x)̂ i+(b_y+c_y)̂ j+(b_z +c_z )̂ k] \)
• \(\vec a.( \vec b+ \vec c)=\) a_x (b_x + c_x) + a_y (b_y + c_y) + a_z (b_z + c_z)
• \(\vec a.( \vec b+ \vec c)=\) ax bx + ax cx + ay by + ay cy + az bz + az cz................................... (1)
• \(\displaystyle \vec a. \vec b+ \vec a.\vec c=(a_x ̂ i+a_y ̂ j+a_z ̂ k).(b_x ̂ i+b_y ̂ j+b_z ̂ k)+(a_x ̂ i+a_y ̂ j+a_z ̂ k).(c_x ̂ i+c_y ̂ j+c_z ̂ k)\)
• \(\displaystyle \vec a. \vec b+ \vec a.\vec c=(a_x.b_x+a_y.b_y +a_z.b_z)+(a_x.c_x +a_y.c_y+a_z.c_z)\)
• \(\vec a. \vec b+ \vec a.\vec c=\) ax bx + ax cx + ay by + ay cy + az bz + az cz................................... (2) 
• From equation (1) and (2)
• ∴ \(\vec a.( \vec b+ \vec c)=\vec a. \vec b+ \vec a.\vec c\)  

Statement II: Cross product over vector addition is distributive

We have to prove \(\vec a × (\vec b + \vec c) = \vec a × \vec b + \vec a × \vec c\)

• \(\displaystyle \vec a\times(\vec b+ \vec c)=(a_x ̂ i+a_y ̂ j+a_z ̂ k)\times[(b_x ̂ i+b_y ̂ j+b_z ̂ k)+(c_x ̂ i+c_y ̂ j+c_z ̂ k)] \)
• ​\(\displaystyle \vec a\times( \vec b+ \vec c)=(a_x ̂ i+a_y ̂ j+a_z ̂ k)\times[(b_x+c_x)̂ i+(b_y+c_y)̂ j+(b_z +c_z )̂ k] \)​
• \(\vec a\times( \vec b+ \vec c)=\begin{bmatrix} \overline i & \overline j & \overline k \\[0.3em] a_x & a_y & a_z \\[0.3em] b_x+c_x &b_y+c_y & b_z+c_z \end{bmatrix}\)
• \(\vec a\times( \vec b+ \vec c)=\) ̂̂î [ay (bz + cz) - az (by + cy)] - ĵ [ax (bz + cz) - az (bx + cx)] + k̂ [ax (by + cy) - ay (bx + cx)] ............(3) 
• \(\displaystyle \vec a\times \vec b+ \vec a\times\vec c=(a_x ̂ i+a_y ̂ j+a_z ̂ k)\times(b_x ̂ i+b_y ̂ j+b_z ̂ k)+(a_x ̂ i+a_y ̂ j+a_z ̂ k)\times(c_x ̂ i+c_y ̂ j+c_z ̂ k)\)
• \(\displaystyle \vec a\times \vec b+ \vec a\times\vec c=\begin{bmatrix} ̂ i & ̂ j & ̂ k \\[0.3em] a_x & a_y & a_z \\[0.3em] b_x &b_y& b_z \end{bmatrix}+\begin{bmatrix} ̂ i & ̂ j & ̂ k \\[0.3em] a_x & a_y & a_z \\[0.3em] c_x &c_y& c_z \end{bmatrix}\)
• \((\vec a\times \vec b+\vec a\times \vec c)=\) ̂̂î [aybz - azby)] - ĵ (axbz - a_zbx)] + k̂ [ax by - aybx] + î [aycz - azcy)] - ĵ (axcz - azcx)] + k̂ [ax cy - aycx]
• \((\vec a\times \vec b+\vec a\times \vec c)=\) î [ay (bz + cz) - az (by + cy)] - ĵ [ax (bz + cz) - az (bx + cx)] + k̂ [ax (by + cy) - ay (bx + cx)] ..............(4) 
• ∴ \(\vec a × (\vec b + \vec c) = \vec a × \vec b + \vec a × \vec c\)

Statement III: Cross product of vectors is associative

• Consider two non-zero perpendicular vectors, a and b.
• We have (a × a) × b = 0 × b = 0
• However, a × b is perpendicular to a and is not the zero vector, so
• a × (a × b) ≠ 0
• (a × a) × b ≠ a × (a × b)
• Cross product of vectors is not associative

∴ Only Statements I and II are correct.