To find a unit vector perpendicular to the triangle defined by point $A(-4, 2, 5)$ and vectors $\mathbf{RAM} = (20, 18, -10)$ and $\mathbf{RAN} = (-10, 8, 15)$, we first need to compute the cross product of the two vectors $\mathbf{RAM}$ and $\mathbf{RAN}$.

Part (a): Unit Vector Perpendicular to the Triangle

The cross product of two vectors gives a vector that is perpendicular to the plane containing the two vectors. Given vectors:

$\mathbf{RAM} = (20, 18, -10), \quad \mathbf{RAN} = (-10, 8, 15)$

Let's compute the cross product $\mathbf{RAM} \times \mathbf{RAN}$.

Step 1: Cross Product Calculation

The cross product of two vectors $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$ is given by:

$\mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2, \, u_3v_1 - u_1v_3, \, u_1v_2 - u_2v_1)$

Substitute the components of $\mathbf{RAM}$ and $\mathbf{RAN}$:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 20 & 18 & -10 \\ -10 & 8 & 15 \end{vmatrix}$$ Calculate each component: $$\mathbf{i}: \quad (18 \cdot 15) - (-10 \cdot 8) = 270 + 80 = 350$$ $$\mathbf{j}: \quad - \left( (20 \cdot 15) - (-10 \cdot (-10)) \right) = - (300 - 100) = -200$$ $$\mathbf{k}: \quad (20 \cdot 8) - (18 \cdot (-10)) = 160 + 180 = 340$$ Thus, the cross product is: $$\mathbf{RAM} \times \mathbf{RAN} = (350, -200, 340)$$ #### Step 2: Compute the Magnitude of the Cross Product To find the unit vector, we first need the magnitude of the cross product: $$\| \mathbf{RAM} \times \mathbf{RAN} \| = \sqrt{350^2 + (-200)^2 + 340^2}$$ Calculate the square of each component: $$350^2 = 122500, \quad (-200)^2 = 40000, \quad 340^2 = 115600$$ $$\| \mathbf{RAM} \times \mathbf{RAN} \| = \sqrt{122500 + 40000 + 115600} = \sqrt{278100}$$ Compute the square root: $$\| \mathbf{RAM} \times \mathbf{RAN} \| = 527.33$$ #### Step 3: Find the Unit Vector Now, divide each component of the cross product by its magnitude to get the unit vector: $$\mathbf{n} = \frac{\mathbf{RAM} \times \mathbf{RAN}}{\| \mathbf{RAM} \times \mathbf{RAN} \|}$$ $$\mathbf{n} = \left( \frac{350}{527.33}, \frac{-200}{527.33}, \frac{340}{527.33} \right)$$ Simplify each component: $$\mathbf{n} \approx \left( 0.664, -0.379, 0.645 \right)$$ ### Answer The unit vector perpendicular to the triangle defined by point $$A$$ and vectors $$\mathbf{RAM}$$ and $$\mathbf{RAN}$$ is: $$\boxed{\mathbf{n} \approx (0.664, -0.379, 0.645)}$$ Would you like more details, or do you have any questions? Here are some related questions you might find interesting: 1. How to compute the angle between two vectors using the dot product? 2. What is the geometric interpretation of the cross product? 3. How do you find the area of a triangle defined by three points in 3D space? 4. What are some applications of cross products in physics and engineering? 5. How would the result change if the vectors were linearly dependent? **Tip:** Always double-check your vector arithmetic when performing cross products, as errors can propagate easily!