Question

If \( \vec{c} \) is a vector along the bisector of the internal angle between the vectors \( \vec{a} = 4\hat{i} + 7\hat{j} - 4\hat{k} \) and \( \vec{b} = 12\hat{i} - 3\hat{j} + 4\hat{k} \), and the magnitude of \( \vec{c} \) is \( 3\sqrt{13} \), then \( \vec{c} = \):

• A. \( 5\hat{i} - 8\hat{j} + 2\sqrt{2} \hat{k} \)
• B. \( 10\hat{i} + 4\hat{j} - \hat{k} \)
• C. \( 7\hat{i} - 10\hat{j} + 4\hat{k} \)
• D. \( 2\sqrt{2} \hat{i} + 5\hat{j} - 8\hat{k} \)

Hint:

To find the angle bisector vector, use the formula \( \vec{c} = \frac{|\vec{b}| \vec{a} + |\vec{a}| \vec{b}}{|\vec{a}| + |\vec{b}|} \). Make sure to adjust the magnitude of the resulting vector to match the given condition.

Answer 1

The Correct Option is B

We are given two vectors \( \vec{a} = 4\hat{i} + 7\hat{j} - 4\hat{k} \) and \( \vec{b} = 12\hat{i} - 3\hat{j} + 4\hat{k} \), and we are asked to find the vector \( \vec{c} \), which lies along the bisector of the internal angle between \( \vec{a} \) and \( \vec{b} \). 
Step 1: The vector \( \vec{c} \) is along the angle bisector, and it can be found using the formula: \[ \vec{c} = \frac{|\vec{b}| \vec{a} + |\vec{a}| \vec{b}}{|\vec{a}| + |\vec{b}|} \] where \( |\vec{a}| \) and \( |\vec{b}| \) are the magnitudes of the vectors \( \vec{a} \) and \( \vec{b} \). First, calculate the magnitudes of \( \vec{a} \) and \( \vec{b} \): \[ |\vec{a}| = \sqrt{4^2 + 7^2 + (-4)^2} = \sqrt{16 + 49 + 16} = \sqrt{81} = 9 \] \[ |\vec{b}| = \sqrt{12^2 + (-3)^2 + 4^2} = \sqrt{144 + 9 + 16} = \sqrt{169} = 13 \] 
Step 2: Now, apply the formula for \( \vec{c} \): \[ \vec{c} = \frac{13(4\hat{i} + 7\hat{j} - 4\hat{k}) + 9(12\hat{i} - 3\hat{j} + 4\hat{k})}{9 + 13} \] \[ \vec{c} = \frac{13(4\hat{i} + 7\hat{j} - 4\hat{k}) + 9(12\hat{i} - 3\hat{j} + 4\hat{k})}{22} \] Simplify each term: \[ \vec{c} = \frac{(52\hat{i} + 91\hat{j} - 52\hat{k}) + (108\hat{i} - 27\hat{j} + 36\hat{k})}{22} \] \[ \vec{c} = \frac{(52\hat{i} + 108\hat{i}) + (91\hat{j} - 27\hat{j}) + (-52\hat{k} + 36\hat{k})}{22} \] \[ \vec{c} = \frac{160\hat{i} + 64\hat{j} - 16\hat{k}}{22} \] \[ \vec{c} = \frac{160}{22} \hat{i} + \frac{64}{22} \hat{j} - \frac{16}{22} \hat{k} \] \[ \vec{c} = \frac{80}{11} \hat{i} + \frac{32}{11} \hat{j} - \frac{8}{11} \hat{k} \] Step 3: Now, we need to adjust the magnitude of \( \vec{c} \) to \( 3\sqrt{13} \). The magnitude of the vector is given as \( 3\sqrt{13} \), and the magnitude of \( \vec{c} \) is found using: \[ |\vec{c}| = \sqrt{\left(\frac{160}{22}\right)^2 + \left(\frac{64}{22}\right)^2 + \left(\frac{16}{22}\right)^2} \] Simplifying this step-by-step calculation will show that the final correct vector is \( 10\hat{i} + 4\hat{j} - \hat{k} \).