Question

problem 12
use the following vectors to answer the questions in parts 1 and 2.
\\(\vec{v} = 2\hat{i} + 3\hat{j} + \hat{k}\\)
\\(\vec{w} = -3\hat{i} + 3\hat{j} + \hat{k}\\)

problem 12 - part 1
(fill in the blank) : calculate the area of the parallelogram with adjacent edges given by \\(\vec{v}\\) and \\(\vec{w}\\).
for full credit, please show the appropriate cross product and other relevant calculations.
area = <your answer here>

problem 12 - part 2
(fill in the blank) : find a unit vector \\(\hat{n}\\) orthogonal to both \\(\vec{v}\\) and \\(\vec{w}\\).
for full credit, please show the relevant calculations. you may use the result from part 1.
\\(\hat{n} = \\) <your answer here>

Explanation:

Step1: Compute cross product $\vec{v} \times \vec{w}$

$$ \vec{v} \times \vec{w} =
LATEXBLOCK0
= \hat{i}(3 \cdot 1 - 1 \cdot 3) - \hat{j}(2 \cdot 1 - 1 \cdot (-3)) + \hat{k}(2 \cdot 3 - 3 \cdot (-3)) = 0\hat{i} - 5\hat{j} + 15\hat{k} $$

Step2: Calculate magnitude for area

$$ \|\vec{v} \times \vec{w}\| = \sqrt{0^2 + (-5)^2 + 15^2} = \sqrt{0 + 25 + 225} = \sqrt{250} = 5\sqrt{10} $$

Step3: Find unit vector for Part 2

$$ \hat{n} = \frac{\vec{v} \times \vec{w}}{\|\vec{v} \times \vec{w}\|} = \frac{0\hat{i} - 5\hat{j} + 15\hat{k}}{5\sqrt{10}} = 0\hat{i} - \frac{1}{\sqrt{10}}\hat{j} + \frac{3}{\sqrt{10}}\hat{k} $$

(Note: $-\hat{n}$ is also a valid orthogonal unit vector)

Answer:

Part 1: $5\sqrt{10}$
Part 2: $-\frac{1}{\sqrt{10}}\hat{j} + \frac{3}{\sqrt{10}}\hat{k}$ (or $\frac{1}{\sqrt{10}}\hat{j} - \frac{3}{\sqrt{10}}\hat{k}$)