Question

If the volume of a parallelepiped whose coterminous edges are $\vec{a}=\hat{i}+\hat{j}+2\,\hat{k}, \vec{b}=2\,\hat{i}+\lambda\,\hat{j}+\hat{k}$ and $\vec{c}=2\,\hat{i}+2\hat{j}+\lambda\hat{k}$ is 35 cu.m, then a value of $\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}-\vec{c}\cdot\vec{a}$ is :

• A. $-10$
• B. $2$
• C. $22$
• D. $-14$

Answer 1

The Correct Option is C

 To find the value of \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} - \vec{c} \cdot \vec{a}\), we start by using the given vectors:

• \(\vec{a} = \hat{i} + \hat{j} + 2\hat{k}\)
• \(\vec{b} = 2\hat{i} + \lambda\hat{j} + \hat{k}\)
• \(\vec{c} = 2\hat{i} + 2\hat{j} + \lambda\hat{k}\)

The volume of the parallelepiped is given by the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 35\).

The cross product \(\vec{b} \times \vec{c}\) is calculated as follows:

	\(\hat{i}\)	\(\hat{j}\)	\(\hat{k}\)
\(b\)	2	\(\lambda\)	1
\(c\)	2	2	\(\lambda\)

Evaluating the determinant gives us:

• \(\vec{b} \times \vec{c} = \left( (\lambda \cdot \lambda - 2 \cdot 1) \hat{i} - (2 \cdot \lambda - 1 \cdot 2) \hat{j} + (2 \cdot 2 - \lambda \cdot 2) \hat{k} \right)\)
• \(= (\lambda^2 - 2) \hat{i} - (2\lambda - 2)\hat{j} + (4 - 2\lambda)\hat{k}\)

Now calculate the dot product \(\vec{a} \cdot (\vec{b} \times \vec{c})\):

• \(\vec{a} \cdot (\vec{b} \times \vec{c}) = (1 \cdot (\lambda^2 - 2)) + (1 \cdot (2 - 2\lambda)) + (2 \cdot (4 - 2\lambda))\)
• \(= \lambda^2 - 2 + 2 - 2\lambda + 8 - 4\lambda\)
• \(= \lambda^2 - 6\lambda + 8\)

Equating this with 35 (given volume), we have:

• \(\lambda^2 - 6\lambda + 8 = 35\)
• \(\lambda^2 - 6\lambda - 27 = 0\)
• Solving this quadratic, we find: \(\lambda = 9\) or \(\lambda = -3\).

To find \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} - \vec{c} \cdot \vec{a}\) with \(\lambda = 9\):

• \(\vec{a} \cdot \vec{b} = 1 \cdot 2 + 1 \cdot 9 + 2 \cdot 1 = 13\)
• \(\vec{b} \cdot \vec{c} = 2 \cdot 2 + 9 \cdot 2 + 1 \cdot 9 = 31\)
• \(\vec{c} \cdot \vec{a} = 2 \cdot 1 + 2 \cdot 1 + 9 \cdot 2 = 22\)

Therefore,

• \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} - \vec{c} \cdot \vec{a} = 13 + 31 - 22 = 22\)

Thus, the answer is \(22\).