Question

For any four vectors a^ vector, b ^ vector, C ^ vector and d ^ vector determine : (a^ vector × b^ vector)(c ^ vector × d ^ vector) + (b^ vector × c^ vector) (a^vector × d^vector) + (c^vector × a^ vector) ( b^vector × d^ vector).

Answer 1

Answer on Question #70004 – Math – Linear Algebra

Question

For any four vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ determine:

$$(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) + (\vec{b} \times \vec{c}) \cdot (\vec{a} \times \vec{d}) + (\vec{c} \times \vec{a}) \cdot (\vec{b} \times \vec{d})$$

Solution

Suppose

$$\vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$$ $$\vec{b} = b_1 \vec{i} + b_2 \vec{j} + b_3 \vec{k}$$ $$\vec{c} = c_1 \vec{i} + c_2 \vec{j} + c_3 \vec{k}$$ $$\vec{d} = d_1 \vec{i} + d_2 \vec{j} + d_3 \vec{k}$$

Then

$$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2 b_3 - a_3 b_2) \vec{i} + (a_3 b_1 - a_1 b_3) \vec{j} + (a_1 b_2 - a_2 b_1) \vec{k}$$ $$\vec{c} \times \vec{d} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ c_1 & c_2 & c_3 \\ d_1 & d_2 & d_3 \end{vmatrix} = (c_2 d_3 - c_3 d_2) \vec{i} + (c_3 d_1 - c_1 d_3) \vec{j} + (c_1 d_2 - c_2 d_1) \vec{k}$$ $$\vec{b} \times \vec{c} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = (b_2 c_3 - b_3 c_2) \vec{i} + (b_3 c_1 - b_1 c_3) \vec{j} + (b_1 c_2 - b_2 c_1) \vec{k}$$ $$\vec{a} \times \vec{d} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ d_1 & d_2 & d_3 \end{vmatrix} = (a_2 d_3 - a_3 d_2) \vec{i} + (a_3 d_1 - a_1 d_3) \vec{j} + (a_1 d_2 - a_2 d_1) \vec{k}$$ $$\vec{c} \times \vec{a} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ c_1 & c_2 & c_3 \\ a_1 & a_2 & a_3 \end{vmatrix} = (c_2 a_3 - c_3 a_2) \vec{i} + (c_3 a_1 - c_1 a_3) \vec{j} + (c_1 a_2 - c_2 a_1) \vec{k}$$ $$\vec{b} \times \vec{d} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ b_1 & b_2 & b_3 \\ d_1 & d_2 & d_3 \end{vmatrix} = (b_2 d_3 - b_3 d_2) \vec{i} + (b_3 d_1 - b_1 d_3) \vec{j} + (b_1 d_2 - b_2 d_1) \vec{k}$$ $$(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (a_2 b_3 - a_3 b_2) (c_2 d_3 - c_3 d_2) +$$ $$\begin{aligned} & + (a_3 b_1 - a_1 b_3) (c_3 d_1 - c_1 d_3) + (a_1 b_2 - a_2 b_1) (c_1 d_2 - c_2 d_1) = \\ & = a_2 b_3 c_2 d_3 - a_3 b_2 c_2 d_3 - a_2 b_3 c_3 d_2 + a_3 b_2 c_3 d_2 + a_3 b_1 c_3 d_1 - a_1 b_3 c_3 d_1 - \end{aligned}$$ $$\begin{array}{l} - a _ {3} b _ {1} c _ {1} d _ {3} + a _ {1} b _ {3} c _ {1} d _ {3} + a _ {1} b _ {2} c _ {1} d _ {2} - a _ {2} b _ {1} c _ {1} d _ {2} - a _ {1} b _ {2} c _ {2} d _ {1} + a _ {2} b _ {1} c _ {2} d _ {1} \\ \left(\vec {b} \times \vec {c}\right) \cdot \left(\vec {a} \times \vec {d}\right) = \left(b _ {2} c _ {3} - b _ {3} c _ {2}\right) \left(a _ {2} d _ {3} - a _ {3} d _ {2}\right) + \\ + \left(b _ {3} c _ {1} - b _ {1} c _ {3}\right) \left(a _ {3} d _ {1} - a _ {1} d _ {3}\right) + \left(b _ {1} c _ {2} - b _ {2} c _ {1}\right) \left(a _ {1} d _ {2} - a _ {2} d _ {1}\right) = \\ = b _ {2} c _ {3} a _ {2} d _ {3} - b _ {3} c _ {2} a _ {2} d _ {3} - b _ {2} c _ {3} a _ {3} d _ {2} + b _ {3} c _ {2} a _ {3} d _ {2} + b _ {3} c _ {1} a _ {3} d _ {1} - b _ {1} c _ {3} a _ {3} d _ {1} - \\ - b _ {3} c _ {1} a _ {1} d _ {3} + b _ {1} c _ {3} a _ {1} d _ {3} + b _ {1} c _ {2} a _ {1} d _ {2} - b _ {2} c _ {1} a _ {1} d _ {2} - b _ {1} c _ {2} a _ {2} d _ {1} + b _ {2} c _ {1} a _ {2} d _ {1} \\ \left(\vec {c} \times \vec {a}\right) \cdot \left(\vec {b} \times \vec {d}\right) = \left(c _ {2} a _ {3} - c _ {3} a _ {2}\right) \left(b _ {2} d _ {3} - b _ {3} d _ {2}\right) + \\ + \left(c _ {3} a _ {1} - c _ {1} a _ {3}\right) \left(b _ {3} d _ {1} - b _ {1} d _ {3}\right) + \left(c _ {1} a _ {2} - c _ {2} a _ {1}\right) \left(b _ {1} d _ {2} - b _ {2} d _ {1}\right) = \\ = c _ {2} a _ {3} b _ {2} d _ {3} - c _ {3} a _ {2} b _ {2} d _ {3} - c _ {2} a _ {3} b _ {3} d _ {2} + c _ {3} a _ {2} b _ {3} d _ {2} + c _ {3} a _ {1} b _ {3} d _ {1} - c _ {1} a _ {3} b _ {3} d _ {1} - \\ - c _ {3} a _ {1} b _ {1} d _ {3} + c _ {1} a _ {3} b _ {1} d _ {3} + c _ {1} a _ {2} b _ {1} d _ {2} - c _ {2} a _ {1} b _ {1} d _ {2} - c _ {1} a _ {2} b _ {2} d _ {1} + c _ {2} a _ {1} b _ {2} d _ {1} \end{array}$$

Hence

$$\begin{array}{l} \left(\vec {a} \times \vec {b}\right) \cdot \left(\vec {c} \times \vec {d}\right) + \left(\vec {b} \times \vec {c}\right) \cdot \left(\vec {a} \times \vec {d}\right) + \left(\vec {c} \times \vec {a}\right) \cdot \left(\vec {b} \times \vec {d}\right) = \\ = a _ {2} b _ {3} c _ {2} d _ {3} - a _ {3} b _ {2} c _ {2} d _ {3} - a _ {2} b _ {3} c _ {3} d _ {2} + a _ {3} b _ {2} c _ {3} d _ {2} + a _ {3} b _ {1} c _ {3} d _ {1} - a _ {1} b _ {3} c _ {3} d _ {1} - \\ - a _ {3} b _ {1} c _ {1} d _ {3} + a _ {1} b _ {3} c _ {1} d _ {3} + a _ {1} b _ {2} c _ {1} d _ {2} - a _ {2} b _ {1} c _ {1} d _ {2} - a _ {1} b _ {2} c _ {2} d _ {1} + a _ {2} b _ {1} c _ {2} d _ {1} + \\ + b _ {2} c _ {3} a _ {2} d _ {3} - b _ {3} c _ {2} a _ {2} d _ {3} - b _ {2} c _ {3} a _ {3} d _ {2} + b _ {3} c _ {2} a _ {3} d _ {2} + b _ {3} c _ {1} a _ {3} d _ {1} - b _ {1} c _ {3} a _ {3} d _ {1} - \\ - b _ {3} c _ {1} a _ {1} d _ {3} + b _ {1} c _ {3} a _ {1} d _ {3} + b _ {1} c _ {2} a _ {1} d _ {2} - b _ {2} c _ {1} a _ {1} d _ {2} - b _ {1} c _ {2} a _ {2} d _ {1} + b _ {2} c _ {1} a _ {2} d _ {1} + \\ + c _ {2} a _ {3} b _ {2} d _ {3} - c _ {3} a _ {2} b _ {2} d _ {3} - c _ {2} a _ {3} b _ {3} d _ {2} + c _ {3} a _ {2} b _ {3} d _ {2} + c _ {3} a _ {1} b _ {3} d _ {1} - c _ {1} a _ {3} b _ {3} d _ {1} - \\ - c _ {3} a _ {1} b _ {1} d _ {3} + c _ {1} a _ {3} b _ {1} d _ {3} + c _ {1} a _ {2} b _ {1} d _ {2} - c _ {2} a _ {1} b _ {1} d _ {2} - c _ {1} a _ {2} b _ {2} d _ {1} + c _ {2} a _ {1} b _ {2} d _ {1} = \\ = a _ {1} \left(- b _ {3} c _ {3} d _ {1} + b _ {3} c _ {1} d _ {3} + b _ {2} c _ {1} d _ {2} - b _ {2} c _ {2} d _ {1} - b _ {3} c _ {1} d _ {3} + b _ {1} c _ {3} d _ {3} + b _ {1} c _ {2} d _ {2} - \right. \\ - b _ {2} c _ {1} d _ {2} + c _ {3} b _ {3} d _ {1} - c _ {3} b _ {1} d _ {3} - c _ {2} b _ {1} d _ {2} + c _ {2} b _ {2} d _ {1}) + a _ {2} \left(b _ {3} c _ {2} d _ {3} - b _ {3} c _ {3} d _ {2} - \right. \\ - b _ {1} c _ {1} d _ {2} + b _ {1} c _ {2} d _ {1} + b _ {2} c _ {3} d _ {3} - b _ {3} c _ {2} d _ {3} - b _ {1} c _ {2} d _ {1} + b _ {2} c _ {1} d _ {1} - c _ {3} b _ {2} d _ {3} + \\ + c _ {3} b _ {3} d _ {2} + c _ {1} b _ {1} d _ {2} - c _ {1} b _ {2} d _ {1}) + a _ {3} \left(- b _ {2} c _ {2} d _ {3} + b _ {2} c _ {3} d _ {2} + b _ {1} c _ {3} d _ {1} - \right. \\ - b _ {1} c _ {1} d _ {3} - b _ {2} c _ {3} d _ {2} + b _ {3} c _ {2} d _ {2} + b _ {3} c _ {1} d _ {1} - b _ {1} c _ {3} d _ {1} + c _ {2} b _ {2} d _ {3} - c _ {2} b _ {3} d _ {2} - \\ - c _ {1} b _ {3} d _ {1} + c _ {1} b _ {1} d _ {3}) = 0 \\ \end{array}$$

Answer: $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) + (\vec{b} \times \vec{c}) \cdot (\vec{a} \times \vec{d}) + (\vec{c} \times \vec{a}) \cdot (\vec{b} \times \vec{d}) = 0.$

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