26. Express $$ (2 \boldsymbol{u}+3 \boldsymbol{v}) imes(4 \boldsymbol{v}-5 \boldsymbol{u}) $$ in terms of $$ (\boldsymbol{u} imes \boldsymbol{v}) $$

Question

26. Express $$ (2 \boldsymbol{u}+3 \boldsymbol{v}) 	imes(4 \boldsymbol{v}-5 \boldsymbol{u}) $$ in terms of $$ (\boldsymbol{u} 	imes \boldsymbol{v}) $$

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We expand the cross product using the distributive property:

$$
(2\boldsymbol{u}+3\boldsymbol{v}) \times (4\boldsymbol{v}-5\boldsymbol{u}) = 2\boldsymbol{u} \times (4\boldsymbol{v}) + 2\boldsymbol{u} \times (-5\boldsymbol{u}) + 3\boldsymbol{v} \times (4\boldsymbol{v}) + 3\boldsymbol{v} \times (-5\boldsymbol{u})
$$

Simplify each term:

$$
= 8\, (\boldsymbol{u} \times \boldsymbol{v}) - 10\, (\boldsymbol{u} \times \boldsymbol{u}) + 12\, (\boldsymbol{v} \times \boldsymbol{v}) - 15\, (\boldsymbol{v} \times \boldsymbol{u})
$$

Recall that for any vector $$\boldsymbol{a}$$, we have $$\boldsymbol{a} \times \boldsymbol{a} = \boldsymbol{0}$$, hence:

$$
\boldsymbol{u} \times \boldsymbol{u} = \boldsymbol{0} \quad \text{and} \quad \boldsymbol{v} \times \boldsymbol{v} = \boldsymbol{0}
$$

Thus, the expression reduces to:

$$
= 8\, (\boldsymbol{u} \times \boldsymbol{v}) - 15\, (\boldsymbol{v} \times \boldsymbol{u})
$$

Next, we apply the anti-commutative property of the cross product:

$$
\boldsymbol{v} \times \boldsymbol{u} = - (\boldsymbol{u} \times \boldsymbol{v})
$$

Substitute this into the expression:

$$
= 8\, (\boldsymbol{u} \times \boldsymbol{v}) - 15\, (- (\boldsymbol{u} \times \boldsymbol{v})) = 8\, (\boldsymbol{u} \times \boldsymbol{v}) + 15\, (\boldsymbol{u} \times \boldsymbol{v})
$$

Combine like terms:

$$
= (8 + 15)\, (\boldsymbol{u} \times \boldsymbol{v}) = 23\, (\boldsymbol{u} \times \boldsymbol{v})
$$

The final result is:

$$
23\, (\boldsymbol{u} \times \boldsymbol{v})
$$
$$ 23\, (\boldsymbol{u} \times \boldsymbol{v}) $$

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