Question

Given the position vectors of points \(A\) and \(B\) as \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \(C\) divides \(AB\) in the ratio 3:2. If \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) is the position vector of point \(D\), find the unit vector in the direction of \( \mathbf{CD} \):

• A. \(\frac{1}{\sqrt{7}}(8\mathbf{i} - 5\mathbf{j} - 3\mathbf{k})\)
• B. \(\frac{1}{\sqrt{266}}(4\mathbf{i} - 13\mathbf{j} + 9\mathbf{k})\)
• C. \(\frac{1}{\sqrt{42}}(8\mathbf{i} - 5\mathbf{j} + 17\mathbf{k})\)
• D. \(\frac{1}{\sqrt{7}}(8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k})\) \vspace{0.5cm}

Hint:

When solving vector problems, make sure to use the section formula to find the position of points dividing a segment and then apply the vector operations correctly to find the direction and magnitude of the resulting vectors.

Answer 1

The Correct Option is C

We are given that the position vectors of points \(A\), \(B\), and \(D\) are \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \), \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \), respectively. We are also told that point \(C\) divides \(AB\) in the ratio \(3:2\). Step 1: Find the position vector of \(C\) using the section formula. \[ \mathbf{C} = \frac{2\mathbf{A} + 3\mathbf{B}}{5} = \frac{2(2\mathbf{i} - 3\mathbf{j} + \mathbf{k}) + 3(\mathbf{i} + 2\mathbf{j} - 3\mathbf{k})}{5} \] \[ \mathbf{C} = \frac{7\mathbf{i} - 7\mathbf{k}}{5} = \frac{7}{5}\mathbf{i} - \frac{7}{5}\mathbf{k} \] Step 2: Find the vector \( \mathbf{CD} \). \[ \mathbf{CD} = \mathbf{D} - \mathbf{C} = (3\mathbf{i} - \mathbf{j} + 2\mathbf{k}) - \left(\frac{7}{5}\mathbf{i} - \frac{7}{5}\mathbf{k}\right) \] \[ \mathbf{CD} = \frac{8}{5}\mathbf{i} - \mathbf{j} + \frac{17}{5}\mathbf{k} \] Step 3: Find the unit vector in the direction of \( \mathbf{CD} \). The magnitude of \( \mathbf{CD} \) is: \[ |\mathbf{CD}| = \sqrt{\left(\frac{8}{5}\right)^2 + (-1)^2 + \left(\frac{17}{5}\right)^2} = \frac{3\sqrt{42}}{5} \] The unit vector in the direction of \( \mathbf{CD} \) is: \[ \hat{\mathbf{CD}} = \frac{1}{\sqrt{42}}(8\mathbf{i} - 5\mathbf{j} + 17\mathbf{k}) \] \vspace{0.5cm}