Question

Let $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}, \, \vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k}$, and a vector $\vec{c}$ be such that $\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k}.$ If $\vec{a} \cdot \vec{c} = 13$, then $(24 - \vec{b} \cdot \vec{c})$ is equal to ______.

Answer 1

From the given equation:

$\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k}.$

Expanding using vector algebra:

$\vec{a} \times \vec{b} + \vec{a} \times \vec{c} + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k}.$

It is given:

$\vec{a} \times \vec{b} = \hat{i} + 8\hat{j} + 13\hat{k}.$

So:

$\vec{a} \times \vec{c} + \vec{b} \times \vec{c} = \vec{0}.$

Expanding further:

$\vec{b} \times \vec{c} = -\vec{a} \times \vec{c}.$

Using $\vec{a} \cdot \vec{c} = 13$, compute:

$\vec{b} \cdot \vec{c} = -\left[\vec{a} \cdot (\hat{i} + 8\hat{j} + 13\hat{k})\right] = -22.$

From the determinant of $\vec{b} \cdot \vec{c}$:

$24 - \vec{b} \cdot \vec{c} = 46.$