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 ai \( +b j+c k \). b. Find the magnitude of \( \overrightarrow{P Q} \). c. Find two unit vectors parallel to \( \overrightarrow{P Q} \). a. Find \( \overrightarrow{P Q} \). \( \overrightarrow{P Q}=\langle 6,2,6\rangle=(6) i+(2) j+(6) k \) b. Find the magnitude of \( \overrightarrow{P Q} \). The magnitude of \( \overrightarrow{P Q} \) is (Type an exact answer, using radicals as needed.)

The magnitude of the vector \( \overrightarrow{P Q} \) can be found using the formula: \[ \|\overrightarrow{P Q}\| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Substituting the coordinates of points \( P \) and \( Q \): \[ \|\overrightarrow{P Q}\| = \sqrt{(7-1)^2 + (7-5)^2 + (11-5)^2} = \sqrt{6^2 + 2^2 + 6^2} = \sqrt{36 + 4 + 36} = \sqrt{76} \] So, the magnitude of \( \overrightarrow{P Q} \) is \( \sqrt{76} \), which can also be simplified to \( 2\sqrt{19} \). To find two unit vectors parallel to \( \overrightarrow{P Q} \), we first need to obtain the unit vector in the direction of \( \overrightarrow{P Q} \). The unit vector can be calculated by dividing the vector by its magnitude: \[ \hat{u} = \frac{\overrightarrow{P Q}}{\|\overrightarrow{P Q}\|} = \frac{\langle 6, 2, 6 \rangle}{\sqrt{76}} = \left\langle \frac{6}{\sqrt{76}}, \frac{2}{\sqrt{76}}, \frac{6}{\sqrt{76}} \right\rangle = \left\langle \frac{3}{\sqrt{19}}, \frac{1}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right\rangle \] The two unit vectors parallel to \( \overrightarrow{P Q} \) are then \( \hat{u} \) and \( -\hat{u} \), which are \( \left\langle \frac{3}{\sqrt{19}}, \frac{1}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right\rangle \) and \( \left\langle -\frac{3}{\sqrt{19}}, -\frac{1}{\sqrt{19}}, -\frac{3}{\sqrt{19}} \right\rangle \).