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 distributiveWe have to prove \(\vec a.( \vec b+ \vec c)=\vec a. \vec b+ \vec a.\vec c\)\(\displaystyle \vec a.( \vec b+ \vec c)=(a_x ? i+a_y ? j+a_z ? k).[(b_x ? i+b_y ? j+b_z ? k)+(c_x ? i+c_y ? j+c_z ? k)] \)\(\displaystyle \vec a.( \vec b+ \vec c)=(a_x ? i+a_y ? j+a_z ? k).[(b_x+c_x)? i+(b_y+c_y)? j+(b_z +c_z )? k] \)\(\vec a.( \vec b+ \vec c)=\) ax (bx + cx) + ay (by + cy) + az (bz + cz)\(\vec a.( \vec b+ \vec c)=\) ax bx + ax cx + ay by + ay cy + az bz + az cz................................... (1)\(\displaystyle \vec a. \vec b+ \vec a.\vec c=(a_x ? i+a_y ? j+a_z ? k).(b_x ? i+b_y ? j+b_z ? k)+(a_x ? i+a_y ? j+a_z ? k).(c_x ? i+c_y ? j+c_z ? k)\)\(\displaystyle \vec a. \vec b+ \vec a.\vec c=(a_x ? i+a_y ? j+a_z ? k).(b_x ? i+b_y ? j+b_z ? k)+(a_x ? i+a_y ? j+a_z ? k).(c_x ? i+c_y ? j+c_z ? k)\) \(\displaystyle \vec a. \vec b+ \vec a.\vec c=(a_x.b_x+a_y.b_y +a_z.b_z)+(a_x.c_x +a_y.c_y+a_z.c_z)\)\(\vec a. \vec b+ \vec a.\vec c=\) 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 distributiveWe have to prove \(\vec a × (\vec b + \vec c) = \vec a × \vec b + \vec a × \vec c\)\(\displaystyle \vec a\times(\vec b+ \vec c)=(a_x ? i+a_y ? j+a_z ? k)\times[(b_x ? i+b_y ? j+b_z ? k)+(c_x ? i+c_y ? j+c_z ? k)] \)?\(\displaystyle \vec a\times( \vec b+ \vec c)=(a_x ? i+a_y ? j+a_z ? k)\times[(b_x+c_x)? i+(b_y+c_y)? j+(b_z +c_z )? k] \)?\(\vec a\times( \vec b+ \vec c)=\begin{bmatrix} \overline i & \overline j & \overline k \\[0.3em] a_x & a_y & a_z \\[0.3em] b_x+c_x &b_y+c_y & b_z+c_z \end{bmatrix}\)\(\vec a\times( \vec b+ \vec c)=\) ??i? [ay (bz + cz) - az (by + cy)] + j? [ax (bz + cz) - az (bx + cx)] + k? [ax (by + cy) - ay (bx + cx)] ............(3) \(\displaystyle \vec a\times \vec b+ \vec a\times\vec c=(a_x ? i+a_y ? j+a_z ? k)\times(b_x ? i+b_y ? j+b_z ? k)+(a_x ? i+a_y ? j+a_z ? k)\times(c_x ? i+c_y ? j+c_z ? k)\)\(\displaystyle \vec a\times \vec b+ \vec a\times\vec c=\begin{bmatrix} ? i & ? j & ? k \\[0.3em] a_x & a_y & a_z \\[0.3em] b_x &b_y& b_z \end{bmatrix}+\begin{bmatrix} ? i & ? j & ? k \\[0.3em] a_x & a_y & a_z \\[0.3em] c_x &c_y& c_z \end{bmatrix}\)\((\vec a\times \vec b+\vec a\times \vec c)=\) ??i? [aybz - azby)] - j? (axbz - azbx)] + k? [ax by - aybx] + i? [aycz - azcy)] - j? (axcz - azcx)] + k? [ax cy - aycx]\((\vec a\times \vec b+\vec a\times \vec c)=\) i? [ay (bz + cz) - az (by + cy)] + j? [ax (bz + cz) - az (bx + cx)] + k? [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 associativeConsider two non-zero perpendicular vectors, a and b.We have (a × a) × b = 0 × b = 0However, a × b is perpendicular to a and is not the zero vector, soa × (a × b) ? 0(a × a) × b ? a × (a × b)Cross product of vectors is not associative? Only Statements I and II are correct.