16. Find the volume of the parallelepiped with $$ \boldsymbol{u}, \boldsymbol{v} $$ and $$ \boldsymbol{w} $$ as adjacent edges. (a) $$ \boldsymbol{u}=\left(\begin{array}{c}3 \ -5 \ 3\end{array} ight), \boldsymbol{v}=\left(\begin{array}{c}1 \ 5 \ -1\end{array} ight), \boldsymbol{w}=\left(\begin{array}{c}3 \ 2 \ -3\end{array} ight) $$ (b) $$ \boldsymbol{u}=4 \boldsymbol{i}+2 \boldsymbol{j}+3 \boldsymbol{k}, \boldsymbol{v}=5 \boldsymbol{i}+6 \boldsymbol{j}+2 \boldsymbol{k}, \boldsymbol{w}=2 \boldsymbol{i}+3 \boldsymbol{j}+5 \boldsymbol{k} $$

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16. Find the volume of the parallelepiped with $$ \boldsymbol{u}, \boldsymbol{v} $$ and $$ \boldsymbol{w} $$ as adjacent edges. (a) $$ \boldsymbol{u}=\left(\begin{array}{c}3 \ -5 \ 3\end{array}ight), \boldsymbol{v}=\left(\begin{array}{c}1 \ 5 \ -1\end{array}ight), \boldsymbol{w}=\left(\begin{array}{c}3 \ 2 \ -3\end{array}ight) $$ (b) $$ \boldsymbol{u}=4 \boldsymbol{i}+2 \boldsymbol{j}+3 \boldsymbol{k}, \boldsymbol{v}=5 \boldsymbol{i}+6 \boldsymbol{j}+2 \boldsymbol{k}, \boldsymbol{w}=2 \boldsymbol{i}+3 \boldsymbol{j}+5 \boldsymbol{k} $$

UpStudy AI · Accepted Answer

**(a) Calculation for** $$ \boldsymbol{u}=\begin{pmatrix}3\ -5\ 3\end{pmatrix} $$, $$ \boldsymbol{v}=\begin{pmatrix}1\ 5\ -1\end{pmatrix} $$, **and** $$ \boldsymbol{w}=\begin{pmatrix}3\ 2\ -3\end{pmatrix} $$

The volume $$ V $$ of a parallelepiped with adjacent edges given by vectors $$ \boldsymbol{u} $$, $$ \boldsymbol{v} $$, and $$ \boldsymbol{w} $$ is given by:
$$
V = \left| \boldsymbol{u} \cdot (\boldsymbol{v} \times \boldsymbol{w}) \right|
$$

1. **Compute** $$ \boldsymbol{v} \times \boldsymbol{w} $$:

Using the formula for the cross product:
   $$
   \boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} v_yw_z - v_zw_y \ v_zw_x - v_xw_z \ v_xw_y - v_yw_x \end{pmatrix}
   $$
   where
   $$
   \boldsymbol{v} = \begin{pmatrix} 1 \ 5 \ -1 \end{pmatrix} \quad \text{and} \quad \boldsymbol{w} = \begin{pmatrix} 3 \ 2 \ -3 \end{pmatrix},
   $$
   we calculate:
   - $$ x\text{-component} $$: 
     $$
     5(-3) - (-1)(2) = -15 + 2 = -13
     $$
   - $$ y\text{-component} $$: 
     $$
     (-1)(3) - (1)(-3) = -3 + 3 = 0
     $$
   - $$ z\text{-component} $$: 
     $$
     (1)(2) - (5)(3) = 2 - 15 = -13
     $$
   So we have:
   $$
   \boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} -13 \ 0 \ -13 \end{pmatrix}
   $$

2. **Compute** $$ \boldsymbol{u} \cdot (\boldsymbol{v} \times \boldsymbol{w}) $$:

With $$ \boldsymbol{u} = \begin{pmatrix} 3 \ -5 \ 3 \end{pmatrix} $$, we compute:
   $$
   \boldsymbol{u} \cdot (\boldsymbol{v} \times \boldsymbol{w}) = 3(-13) + (-5)(0) + 3(-13) = -39 + 0 - 39 = -78
   $$

3. **Take the absolute value to find the volume**:
   $$
   V = \left| -78 \right| = 78
   $$

---

**(b) Calculation for** $$ \boldsymbol{u}=4\boldsymbol{i}+2\boldsymbol{j}+3\boldsymbol{k} $$, $$ \boldsymbol{v}=5\boldsymbol{i}+6\boldsymbol{j}+2\boldsymbol{k} $$, **and** $$ \boldsymbol{w}=2\boldsymbol{i}+3\boldsymbol{j}+5\boldsymbol{k} $$

1. **Express the vectors in component form**:
   $$
   \boldsymbol{u} = \begin{pmatrix} 4 \ 2 \ 3 \end{pmatrix}, \quad \boldsymbol{v} = \begin{pmatrix} 5 \ 6 \ 2 \end{pmatrix}, \quad \boldsymbol{w} = \begin{pmatrix} 2 \ 3 \ 5 \end{pmatrix}
   $$

2. **Compute** $$ \boldsymbol{v} \times \boldsymbol{w} $$:

Using the cross product formula:
   $$
   \boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} v_yw_z - v_zw_y \ v_zw_x - v_xw_z \ v_xw_y - v_yw_x \end{pmatrix}
   $$
   where
   $$
   \boldsymbol{v} = \begin{pmatrix} 5 \ 6 \ 2 \end{pmatrix} \quad \text{and} \quad \boldsymbol{w} = \begin{pmatrix} 2 \ 3 \ 5 \end{pmatrix},
   $$
   we calculate:
   - $$ x\text{-component} $$: 
     $$
     6(5) - 2(3) = 30 - 6 = 24
     $$
   - $$ y\text{-component} $$: 
     $$
     2(2) - 5(5) = 4 - 25 = -21
     $$
   - $$ z\text{-component} $$: 
     $$
     5(3) - 6(2) = 15 - 12 = 3
     $$
   So we have:
   $$
   \boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} 24 \ -21 \ 3 \end{pmatrix}
   $$

3. **Compute** $$ \boldsymbol{u} \cdot (\boldsymbol{v} \times \boldsymbol{w}) $$:

With $$ \boldsymbol{u} = \begin{pmatrix} 4 \ 2 \ 3 \end{pmatrix} $$, we compute:
   $$
   \boldsymbol{u} \cdot (\boldsymbol{v} \times \boldsymbol{w}) = 4(24) + 2(-21) + 3(3) = 96 - 42 + 9 = 63
   $$

4. **Take the absolute value to find the volume**:
   $$
   V = \left| 63 \right| = 63
   $$

---

**Final Answers:**

- (a) The volume is $$ 78 $$.
- (b) The volume is $$ 63 $$.
The volume of the parallelepiped for part (a) is 78, and for part (b) is 63.

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