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Step-by-Step Solution
Step 1: Represent the vectors in component form
Let
\overrightarrow{b} = \hat{i} + \hat{j} + \lambda \,\hat{k},
\quad
\overrightarrow{a} = x \,\hat{i} + y \,\hat{j} + z \,\hat{k}.
Step 2: Use the given cross product condition
We are given
\overrightarrow{a} \times \overrightarrow{b} = 13 \,\hat{i} \;-\; \hat{j} \;-\; 4 \,\hat{k}.
Recall that
\overrightarrow{a} \times \overrightarrow{b}
\;=\;
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\[6pt]
x & y & z \\[6pt]
1 & 1 & \lambda
\end{vmatrix}
\;=\;
\hat{i}(\lambda y - z)
+ \hat{j}\,(z - \lambda x)
+ \hat{k}\,(x - y).
By comparing components with 13\,\hat{i} - \hat{j} - 4\,\hat{k} , we obtain:
\lambda y - z = 13, \quad
z - \lambda x = -1, \quad
x - y = -4.
Step 3: Use the given dot product condition
We also know
\overrightarrow{a} \cdot \overrightarrow{b} + 21 = 0,
which implies
\overrightarrow{a} \cdot \overrightarrow{b} = x \cdot 1 + y \cdot 1 + z \cdot \lambda = -21.
Hence,
x + y + \lambda z = -21.
Step 4: Solve the system of equations
From the cross product equations and the dot product equation, we have four equations:
\lambda y - z = 13
z - \lambda x = -1
x - y = -4
x + y + \lambda z = -21
By suitable manipulation (or by inspection), one finds that \lambda = 3 . Substituting \lambda = 3 back into the equations yields:
3y - z = 13
z - 3x = -1
x - y = -4
x + y + 3z = -21
Solving consistently gives:
x = -2,
\quad
y = 2,
\quad
z = -7.
Step 5: Determine the required vectors
Now that x, y, z, and \lambda are known, calculate:
\overrightarrow{b} - \overrightarrow{a}
= (1 - x)\,\hat{i} \;+\; (1 - y)\,\hat{j} \;+\; (\lambda - z)\,\hat{k}
= (1 - (-2))\,\hat{i} \;+\; (1 - 2)\,\hat{j} \;+\; (3 - (-7))\,\hat{k}
= 3\,\hat{i} \;-\;\hat{j} \;+\;10\,\hat{k}.
\overrightarrow{b} + \overrightarrow{a}
= (1 + x)\,\hat{i} \;+\; (1 + y)\,\hat{j} \;+\; (\lambda + z)\,\hat{k}
= (1 + (-2))\,\hat{i} \;+\; (1 + 2)\,\hat{j} \;+\; (3 + (-7))\,\hat{k}
= -\,\hat{i} \;+\;3\,\hat{j} \;-\;4\,\hat{k}.
Step 6: Compute the dot products
We need to evaluate
\bigl(\overrightarrow{b} - \overrightarrow{a}\bigr)
\cdot
(\hat{k} - \hat{j})
\;+\;
\bigl(\overrightarrow{b} + \overrightarrow{a}\bigr)
\cdot
(\hat{i} - \hat{k}).
\hat{k} - \hat{j} = (0, -1, 1).
Hence,
(\overrightarrow{b} - \overrightarrow{a}) \cdot (\hat{k} - \hat{j})
\;=\;
(3, -1, 10) \cdot (0, -1, 1)
= 3 \cdot 0 + (-1) \cdot (-1) + 10 \cdot 1
= 1 + 10
= 11.
\hat{i} - \hat{k} = (1, 0, -1).
Hence,
(\overrightarrow{b} + \overrightarrow{a}) \cdot (\hat{i} - \hat{k})
\;=\;
(-1, 3, -4) \cdot (1, 0, -1)
= (-1) \cdot 1 + 3 \cdot 0 + (-4) \cdot (-1)
= -1 + 4
= 3.
Step 7: Summation to find the final result
Adding these two dot products gives:
11 \;+\; 3 = 14.
Therefore, the required value is \boxed{14} .
