EXERCISE 5.1 1. Find the volume of the parallelopiped with adjacent sides. $$ \overline{O A}=3 \hat{i}-\hat{j}, \quad \overline{O B}=\hat{j}+2 \hat{k} \text {, and } \overline{O C}=\hat{i}+5 \hat{j}+4 \hat{k} $$ extending from the origin of co-ordinates $$ O $$

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To find the volume of the parallelepiped defined by the vectors $$\overline{OA}$$, $$\overline{OB}$$, and $$\overline{OC}$$, we can use the **scalar triple product**. The volume $$V$$ is given by the absolute value of the scalar triple product of the three vectors:

$$
V = \left| \overline{OA} \cdot (\overline{OB} \times \overline{OC}) \right|
$$

Let's proceed step by step.

### Step 1: Express the Vectors in Component Form

Given:
$$
\overline{OA} = 3\hat{i} - \hat{j} = \begin{pmatrix} 3 \ -1 \ 0 \end{pmatrix}
$$
$$
\overline{OB} = \hat{j} + 2\hat{k} = \begin{pmatrix} 0 \ 1 \ 2 \end{pmatrix}
$$
$$
\overline{OC} = \hat{i} + 5\hat{j} + 4\hat{k} = \begin{pmatrix} 1 \ 5 \ 4 \end{pmatrix}
$$

### Step 2: Compute the Cross Product $$\overline{OB} \times \overline{OC}$$

The cross product of two vectors $$\mathbf{B} = \begin{pmatrix} B_x \ B_y \ B_z \end{pmatrix}$$ and $$\mathbf{C} = \begin{pmatrix} C_x \ C_y \ C_z \end{pmatrix}$$ is:

$$
\mathbf{B} \times \mathbf{C} = \begin{pmatrix}
B_y C_z - B_z C_y \
B_z C_x - B_x C_z \
B_x C_y - B_y C_x
\end{pmatrix}
$$

Applying this to $$\overline{OB}$$ and $$\overline{OC}$$:

$$
\overline{OB} \times \overline{OC} = \begin{pmatrix}
1 \cdot 4 - 2 \cdot 5 \
2 \cdot 1 - 0 \cdot 4 \
0 \cdot 5 - 1 \cdot 1
\end{pmatrix} = \begin{pmatrix}
4 - 10 \
2 - 0 \
0 - 1
\end{pmatrix} = \begin{pmatrix}
-6 \
2 \
-1
\end{pmatrix}
$$

### Step 3: Compute the Scalar Triple Product

Now, take the dot product of $$\overline{OA}$$ with $$\overline{OB} \times \overline{OC}$$:

$$
\overline{OA} \cdot (\overline{OB} \times \overline{OC}) = \begin{pmatrix} 3 \ -1 \ 0 \end{pmatrix} \cdot \begin{pmatrix} -6 \ 2 \ -1 \end{pmatrix} = (3)(-6) + (-1)(2) + (0)(-1) = -18 - 2 + 0 = -20
$$

### Step 4: Determine the Volume

The volume is the absolute value of the scalar triple product:

$$
V = \left| -20 \right| = 20 \text{ cubic units}
$$

### Conclusion

**The volume of the parallelepiped is 20 cubic units.**
The volume of the parallelepiped is 20 cubic units.

EXERCISE 5.1 1. Find the volume of the parallelopiped with adjacent sides. \[ \overline{O A}=3 \hat{i}-\hat{j}, \quad \overline{O B}=\hat{j}+2 \hat{k} \text {, and } \overline{O C}=\hat{i}+5 \hat{j}+4 \hat{k} \] extending from the origin of co-ordinates \( O \)

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