Key Concepts and Formulas

• Volume of a Parallelepiped: The volume of a parallelepiped with coterminous edges given by vectors $\overrightarrow a$, $\overrightarrow b$, and $\overrightarrow c$ is the absolute value of their scalar triple product, i.e., $V = |[\overrightarrow a \overrightarrow b \overrightarrow c]| = |\overrightarrow a \cdot (\overrightarrow b \times \overrightarrow c)|$.
• Scalar Triple Product as a Determinant: The scalar triple product can be computed as the determinant of the matrix formed by the components of the three vectors. If $\overrightarrow a = a_1 \widehat i + a_2 \widehat j + a_3 \widehat k$, $\overrightarrow b = b_1 \widehat i + b_2 \widehat j + b_3 \widehat k$, and $\overrightarrow c = c_1 \widehat i + c_2 \widehat j + c_3 \widehat k$, then $[\overrightarrow a \overrightarrow b \overrightarrow c] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$.
• Dot Product of Vectors: For two vectors $\overrightarrow p = p_1 \widehat i + p_2 \widehat j + p_3 \widehat k$ and $\overrightarrow q = q_1 \widehat i + q_2 \widehat j + q_3 \widehat k$, their dot product is $\overrightarrow p \cdot \overrightarrow q = p_1 q_1 + p_2 q_2 + p_3 q_3$.

Step-by-Step Solution

Step 1: Set up the scalar triple product determinant for the volume. We are given the coterminus edges of the parallelepiped as vectors: $\overrightarrow a = \widehat i + \widehat j + n\widehat k$ $\overrightarrow b = 2\widehat i + 4\widehat j - n\widehat k$ $\overrightarrow c = \widehat i + n\widehat j + 3\widehat k$ The volume $V$ of the parallelepiped is given as 158 cubic units. The volume is the absolute value of the scalar triple product: $V = \left| \begin{vmatrix} 1 & 1 & n \\ 2 & 4 & -n \\ 1 & n & 3 \end{vmatrix} \right|$ Given $V = 158$, we have: $158 = \left| \begin{vmatrix} 1 & 1 & n \\ 2 & 4 & -n \\ 1 & n & 3 \end{vmatrix} \right|$

Step 2: Expand the determinant. We expand the determinant along the first row: $\begin{vmatrix} 1 & 1 & n \\ 2 & 4 & -n \\ 1 & n & 3 \end{vmatrix} = 1 \cdot \begin{vmatrix} 4 & -n \\ n & 3 \end{vmatrix} - 1 \cdot \begin{vmatrix} 2 & -n \\ 1 & 3 \end{vmatrix} + n \cdot \begin{vmatrix} 2 & 4 \\ 1 & n \end{vmatrix}$ Calculate the $2 \times 2$ determinants: $\begin{vmatrix} 4 & -n \\ n & 3 \end{vmatrix} = (4)(3) - (-n)(n) = 12 + n^2$ $\begin{vmatrix} 2 & -n \\ 1 & 3 \end{vmatrix} = (2)(3) - (-n)(1) = 6 + n$ $\begin{vmatrix} 2 & 4 \\ 1 & n \end{vmatrix} = (2)(n) - (4)(1) = 2n - 4$ Substitute these back into the expansion: $\text{Determinant} = 1(12 + n^2) - 1(6 + n) + n(2n - 4)$ $\text{Determinant} = 12 + n^2 - 6 - n + 2n^2 - 4n$

Step 3: Simplify the determinant expression and form an equation. Combine like terms to simplify the determinant: $\text{Determinant} = (n^2 + 2n^2) + (-n - 4n) + (12 - 6)$ $\text{Determinant} = 3n^2 - 5n + 6$ Now, we equate the absolute value of this determinant to the given volume: $|3n^2 - 5n + 6| = 158$ This gives two possibilities:

1. $3n^2 - 5n + 6 = 158$
2. $3n^2 - 5n + 6 = -158$

Step 4: Solve the first equation for $n$. $3n^2 - 5n + 6 = 158$ $3n^2 - 5n + 6 - 158 = 0$ $3n^2 - 5n - 152 = 0$ We can factor this quadratic equation. We look for two numbers that multiply to $3 \times (-152) = -456$ and add up to $-5$. These numbers are $-24$ and $19$. $3n^2 - 24n + 19n - 152 = 0$ Factor by grouping: $3n(n - 8) + 19(n - 8) = 0$ $(3n + 19)(n - 8) = 0$ This yields two possible values for $n$: $3n + 19 = 0 \Rightarrow n = -\frac{19}{3}$ $n - 8 = 0 \Rightarrow n = 8$

Step 5: Solve the second equation for $n$. $3n^2 - 5n + 6 = -158$ $3n^2 - 5n + 6 + 158 = 0$ $3n^2 - 5n + 164 = 0$ To check for real solutions, we can calculate the discriminant ($\Delta = b^2 - 4ac$): $\Delta = (-5)^2 - 4(3)(164) = 25 - 12(164) = 25 - 1968 = -1943$ Since the discriminant is negative, this quadratic equation has no real solutions for $n$.

Step 6: Apply the constraint $n \ge 0$ to find the valid value of $n$. From Step 4, we obtained $n = -\frac{19}{3}$ and $n = 8$. The problem states that $n \ge 0$. Therefore, $n = -\frac{19}{3}$ is rejected, and the valid value is $n = 8$.

Step 7: Evaluate the options using the determined value of $n=8$. The vectors are: $\overrightarrow a = \widehat i + \widehat j + 8\widehat k$ $\overrightarrow b = 2\widehat i + 4\widehat j - 8\widehat k$ $\overrightarrow c = \widehat i + 8\widehat j + 3\widehat k$

Option (A): $n = 7$. This is incorrect as we found $n=8$. Option (D): $n = 9$. This is incorrect as we found $n=8$.

Option (B): $\overrightarrow b \cdot \overrightarrow c$ $\overrightarrow b \cdot \overrightarrow c = (2)(\widehat i) \cdot (1)(\widehat i) + (4)(\widehat j) \cdot (8)(\widehat j) + (-8)(\widehat k) \cdot (3)(\widehat k)$ $\overrightarrow b \cdot \overrightarrow c = (2)(1) + (4)(8) + (-8)(3)$ $\overrightarrow b \cdot \overrightarrow c = 2 + 32 - 24 = 10$ So, $\overrightarrow b \cdot \overrightarrow c = 10$. This matches option (B).

Option (C): $\overrightarrow a \cdot \overrightarrow c$ $\overrightarrow a \cdot \overrightarrow c = (1)(\widehat i) \cdot (1)(\widehat i) + (1)(\widehat j) \cdot (8)(\widehat j) + (8)(\widehat k) \cdot (3)(\widehat k)$ $\overrightarrow a \cdot \overrightarrow c = (1)(1) + (1)(8) + (8)(3)$ $\overrightarrow a \cdot \overrightarrow c = 1 + 8 + 24 = 33$ So, $\overrightarrow a \cdot \overrightarrow c = 33$. This does not match option (C).

The only correct option is (B).

Common Mistakes & Tips

• Sign Errors in Determinant Expansion: Carefully apply the alternating signs (+, -, +) when expanding the determinant. A single sign error can lead to an incorrect quadratic equation.
• Forgetting Absolute Value: The volume is always non-negative. Ensure you consider both positive and negative values for the determinant if the parameter's range allows, or use the absolute value at the end. In this case, solving $|X|=158$ leads to $X=158$ or $X=-158$.
• Ignoring Constraints: Always check if the calculated values of the variable satisfy the given constraints (e.g., $n \ge 0$). This is crucial for selecting the correct solution.

Summary

The volume of a parallelepiped is found using the scalar triple product, which can be calculated as the determinant of the matrix formed by the vectors. By setting the absolute value of this determinant equal to the given volume (158), we formed a quadratic equation in $n$. Solving this quadratic equation and applying the constraint $n \ge 0$ yielded $n=8$. Subsequently, we evaluated the dot products in options (B) and (C) with $n=8$ to determine the correct answer.

The final answer is \boxed{A}.