Given $$ { }^{\prime} \vec{u} imes \vec{v}=\langle 1,-2,3 angle $$, find $$ (\vec{u}-2 \vec{v}) imes(\vec{u}+5 \vec{v}) $$

Question

Given $$ { }^{\prime} \vec{u} 	imes \vec{v}=\langle 1,-2,3angle $$, find $$ (\vec{u}-2 \vec{v}) 	imes(\vec{u}+5 \vec{v}) $$

UpStudy AI · Accepted Answer

To find $$(\vec{u} - 2\vec{v}) \times (\vec{u} + 5\vec{v})$$ given that $$\vec{u} \times \vec{v} = \langle 1, -2, 3 \rangle$$, we can use the properties of the cross product.

### Step-by-Step Solution:

1. **Expand the Cross Product:**
   $$
   (\vec{u} - 2\vec{v}) \times (\vec{u} + 5\vec{v}) = \vec{u} \times \vec{u} + 5\vec{u} \times \vec{v} - 2\vec{v} \times \vec{u} - 10\vec{v} \times \vec{v}
   $$

2. **Simplify Using Cross Product Properties:**
   - $$\vec{a} \times \vec{a} = \vec{0}$$ for any vector $$\vec{a}$$.
   - $$\vec{v} \times \vec{u} = -\vec{u} \times \vec{v}$$.

Applying these:
   $$
   \vec{u} \times \vec{u} = \vec{0}, \quad \vec{v} \times \vec{v} = \vec{0}
   $$
   $$
   -2\vec{v} \times \vec{u} = 2\vec{u} \times \vec{v}
   $$

So the expression simplifies to:
   $$
   0 + 5\vec{u} \times \vec{v} + 2\vec{u} \times \vec{v} + 0 = 7\vec{u} \times \vec{v}
   $$

3. **Substitute the Given Cross Product:**
   $$
   7\vec{u} \times \vec{v} = 7 \langle 1, -2, 3 \rangle = \langle 7, -14, 21 \rangle
   $$

### Final Answer:
$$
(\vec{u} - 2\vec{v}) \times (\vec{u} + 5\vec{v}) = \langle 7,\, -14,\, 21 \rangle
$$
$$ (\vec{u} - 2\vec{v}) \times (\vec{u} + 5\vec{v}) = \langle 7, -14, 21 \rangle $$

Given , find

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