Question

If $\vec{a} = 2\vec{i} + 3\vec{j} - \vec{k}$, $\vec{b} = -\vec{i} + 2\vec{j} - 4\vec{k}$ and $\vec{c} = \vec{i} + \vec{j} + \vec{k}$, then $(\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c}) =$

• A. $-74$
• B. $74$
• C. $-42$
• D. $42$

Hint:

Using Lagrange's identity for cross products saves significant time compared to computing the cross products $\vec{a}\times\vec{b}$ and $\vec{a}\times\vec{c}$ individually.

Answer 1

The Correct Option is A

Step 1: Concept
We use the vector identity for the dot product of two cross products: \[ (\vec{u} \times \vec{v}) \cdot (\vec{w} \times \vec{z}) = (\vec{u} \cdot \vec{w})(\vec{v} \cdot \vec{z}) - (\vec{u} \cdot \vec{z})(\vec{v} \cdot \vec{w}) \]

Step 2: Meaning
Substituting $\vec{u} = \vec{a}$, $\vec{v} = \vec{b}$, $\vec{w} = \vec{a}$, and $\vec{z} = \vec{c}$ into the identity, we get: \[ (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c}) = (\vec{a} \cdot \vec{a})(\vec{b} \cdot \vec{c}) - (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{a}) \]

Step 3: Analysis
First, let us calculate the individual dot products:
• $\vec{a} \cdot \vec{a} = 2^2 + 3^2 + (-1)^2 = 4 + 9 + 1 = 14$
• $\vec{b} \cdot \vec{c} = (-1)(1) + (2)(1) + (-4)(1) = -1 + 2 - 4 = -3$
• $\vec{a} \cdot \vec{c} = (2)(1) + (3)(1) + (-1)(1) = 2 + 3 - 1 = 4$
• $\vec{b} \cdot \vec{a} = (-1)(2) + (2)(3) + (-4)(-1) = -2 + 6 + 4 = 8$ Now, substitute these values back into the expression: \[ (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c}) = (14)(-3) - (4)(8) \] \[ \implies -42 - 32 = -74 \]

Step 4: Conclusion
The value of $(\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c})$ is $-74$. Final Answer: (A)