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

Correct Answer: 46

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 \( \vec{b} \cdot \vec{c} \):

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