Question

The volume of the parallelepiped whose coterminous vectors are given by the vectors \[ \overrightarrow{a} = i - j + k, \quad \overrightarrow{b} = 3i + j - k, \quad \overrightarrow{c} = 5i + 2j - 7k \] is (in cubic units):

• A. 15
• B. 20
• C. 16
• D. 18
• E. 22

Hint:

The volume of a parallelepiped can be found using the scalar triple product \( \left| \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \right| \). Remember to compute the cross product first and then the dot product.

Answer 1

The Correct Option is B

The volume of the parallelepiped formed by three vectors \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) is given by the scalar triple product:

\[ V = \left| \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \right|. \]
Step 1: Compute the cross product \( \overrightarrow{b} \times \overrightarrow{c} \):

\[ \overrightarrow{b} = 3i + j - k, \quad \overrightarrow{c} = 5i + 2j - 7k. \] The cross product is:

\[ \overrightarrow{b} \times \overrightarrow{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & -1 \\ 5 & 2 & -7 \end{vmatrix}. \] Using the determinant formula:

\[ = \hat{i} \begin{vmatrix} 1 & -1 \\ 2 & -7 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & -1 \\ 5 & -7 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 1 \\ 5 & 2 \end{vmatrix}. \] This simplifies to:

\[ = \hat{i} \left( 1(-7) - (-1)(2) \right) - \hat{j} \left( 3(-7) - (-1)(5) \right) + \hat{k} \left( 3(2) - 1(5) \right) \] \[ = \hat{i} \left( -7 + 2 \right) - \hat{j} \left( -21 + 5 \right) + \hat{k} \left( 6 - 5 \right) \] \[ = -5\hat{i} + 16\hat{j} + \hat{k}. \] Thus, \[ \overrightarrow{b} \times \overrightarrow{c} = -5\hat{i} + 16\hat{j} + \hat{k}. \] Step 2: Compute the dot product \( \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \):

\[ \overrightarrow{a} = i - j + k. \] Now, take the dot product:

\[ \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) = (i - j + k) \cdot (-5\hat{i} + 16\hat{j} + \hat{k}). \] This gives:

\[ = 1(-5) + (-1)(16) + 1(1) = -5 - 16 + 1 = -20. \] Step 3: The volume is the absolute value of this result:

\[ V = \left| -20 \right| = 20. \]
Thus, the volume of the parallelepiped is 20 cubic units.

Therefore, the correct answer is option (B).