Question

Find a unit vector in the direction of \( \overrightarrow{DA} \).

Hint:

To find a unit vector, divide the vector by its magnitude.

Answer 1

Step 1: Compute \( \overrightarrow{DA} \)
\[ \overrightarrow{DA} = \text{Position vector of } A - \text{Position vector of } D \] \[ \overrightarrow{DA} = (7\hat{i} + 5\hat{j} + 8\hat{k}) - (2\hat{i} + 3\hat{j} + 4\hat{k}) = 5\hat{i} + 2\hat{j} + 4\hat{k}. \] Step 2: Find the magnitude of \( \overrightarrow{DA} \)
\[ |\overrightarrow{DA}| = \sqrt{(5)^2 + (2)^2 + (4)^2} = \sqrt{25 + 4 + 16} = \sqrt{45} = 3\sqrt{5}. \] Step 3: Compute the unit vector
The unit vector is: \[ \hat{u} = \frac{\overrightarrow{DA}}{|\overrightarrow{DA}|} = \frac{5\hat{i} + 2\hat{j} + 4\hat{k}}{3\sqrt{5}} = \frac{5}{3\sqrt{5}}\hat{i} + \frac{2}{3\sqrt{5}}\hat{j} + \frac{4}{3\sqrt{5}}\hat{k}. \] Step 4: Final result
The unit vector in the direction of \( \overrightarrow{DA} \) is: \[ \frac{5}{3\sqrt{5}}\hat{i} + \frac{2}{3\sqrt{5}}\hat{j} + \frac{4}{3\sqrt{5}}\hat{k}. \]