Question

Three vectors \(\hat{\mathrm{i}}-\hat{\mathrm{k}}, \lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+(1-\lambda) \hat{\mathrm{k}}\) and \(\mu \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+(1+\lambda-\mu) \hat{\mathrm{k}}\) represents coterminous edges of a parallelopiped, then the volume of the parallelopiped depends on.

• A. only \(\lambda\)
• B. only \(\mu\)
• C. both \(\lambda\) and \(\mu\)
• D. neither \(\lambda\) nor \(\mu\)

Hint:

The volume is the absolute value of the scalar triple product, which is the determinant of the three vectors.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Volume of a parallelepiped \(=|[\bar{a}\ \bar{b}\ \bar{c}]|\), the absolute value of the determinant of the components.

Step 2: Write the Determinant
Rows: \((1,0,-1)\), \((\lambda,1,1-\lambda)\), \((\mu,\lambda,1+\lambda-\mu)\).

Step 3: Expand
Determinant \(=1\left[(1+\lambda-\mu)-\lambda(1-\lambda)\right]-1\left[\lambda^{2}-\mu\right]\).
\(=(1-\mu+\lambda^{2})-\lambda^{2}+\mu=1\).

Step 4: Conclusion
The volume is 1 whatever \(\lambda\) and \(\mu\) are, so it depends on neither, option (D).

Final Answer:
The volume is always 1, so it depends on neither \(\lambda\) nor \(\mu\). \[ \boxed{\text{(D) }\text{neither }\lambda\text{ nor }\mu} \]