Consider the points \( P(1,5,5) \) and \( Q(7,7,11) \). a. Find \( \overrightarrow{P Q} \) and state your answer in two forms: \( \langle a, b, c\rangle \) and \( a i+b j+c k \). b. Find the magnitude of \( \overrightarrow{P Q} \). c. Find two unit vectors parallel to \( \overrightarrow{P Q} \). a. Find \( \overrightarrow{P Q} \). \( \overrightarrow{P Q}=\langle 6,2,6\rangle=(6) i+(2) j+(6) \mathbf{k} \) b. Find the magnitude of \( \overrightarrow{P Q} \). The magnitude of \( \overrightarrow{P Q} \) is \( 2 \sqrt{19} \). (Type an exact answer, using radicals as needed.) c. Find two unit vectors parallel to \( \overrightarrow{P Q} \). The unit vector in the direction of \( \overrightarrow{P Q} \) is \( \langle\square, \square, \square \). (Type exact answers, using radicals as needed.)

**c. Find two unit vectors parallel to \( \overrightarrow{P Q} \).** To find unit vectors parallel to \( \overrightarrow{P Q} \), we first determine the unit vector in the direction of \( \overrightarrow{P Q} \) and then consider its opposite direction. 1. **Unit Vector in the Direction of \( \overrightarrow{P Q} \):** Given: \[ \overrightarrow{P Q} = \langle 6, 2, 6 \rangle \] and its magnitude: \[ \|\overrightarrow{P Q}\| = 2\sqrt{19} \] The unit vector \( \mathbf{u} \) in the direction of \( \overrightarrow{P Q} \) is: \[ \mathbf{u} = \frac{\overrightarrow{P Q}}{\|\overrightarrow{P Q}\|} = \left\langle \frac{6}{2\sqrt{19}}, \frac{2}{2\sqrt{19}}, \frac{6}{2\sqrt{19}} \right\rangle = \left\langle \frac{3}{\sqrt{19}}, \frac{1}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right\rangle \] In component form: \[ \mathbf{u} = \frac{3}{\sqrt{19}}\, \mathbf{i} + \frac{1}{\sqrt{19}}\, \mathbf{j} + \frac{3}{\sqrt{19}}\, \mathbf{k} \] 2. **Unit Vector in the Opposite Direction:** The opposite unit vector \( \mathbf{v} \) is simply the negative of \( \mathbf{u} \): \[ \mathbf{v} = -\mathbf{u} = \left\langle -\frac{3}{\sqrt{19}}, -\frac{1}{\sqrt{19}}, -\frac{3}{\sqrt{19}} \right\rangle \] In component form: \[ \mathbf{v} = -\frac{3}{\sqrt{19}}\, \mathbf{i} - \frac{1}{\sqrt{19}}\, \mathbf{j} - \frac{3}{\sqrt{19}}\, \mathbf{k} \] **Final Answer:** Two unit vectors parallel to \( \overrightarrow{P Q} \) are \[ \left\langle\, \frac{3}{\sqrt{19}},\ \frac{1}{\sqrt{19}},\ \frac{3}{\sqrt{19}}\,\right\rangle \] and \[ \left\langle\, -\frac{3}{\sqrt{19}},\ -\frac{1}{\sqrt{19}},\ -\frac{3}{\sqrt{19}}\,\right\rangle. \]