Problem 25 - Precalculus Mathematics for Calculus

Question

Find the area of \(\triangle P Q R\). $$P(1,0,1), Q(0,1,0), R(2,3,4)$$

Step-by-Step Solution

Answer

The area of \(\triangle PQR\) is \(\sqrt{14}\).

Step 1: Write the formula for the area of a triangle in 3D

The area of a triangle with vertices at points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), and \((x_3, y_3, z_3)\) is given by the formula: \[\text{Area} = \frac{1}{2} \sqrt{\left( y_2 - y_1 \right)\left( z_3 - z_1 \right) - \left( z_2 - z_1 \right)\left( y_3 - y_1 \right)^2 + \left( z_2 - z_1 \right)\left( x_3 - x_1 \right) - \left( x_2 - x_1 \right)\left( z_3 - z_1 \right)^2 + \left( x_2 - x_1 \right)\left( y_3 - y_1 \right) - \left( y_2 - y_1 \right)\left( x_3 - x_1 \right)^2}\].

Step 2: Identify the coordinates of the points

We have the points provided as follows: \( P(1,0,1) \), \( Q(0,1,0) \), and \( R(2,3,4) \). Identify \((x_1, y_1, z_1) = (1,0,1)\), \((x_2, y_2, z_2) = (0,1,0)\), and \((x_3, y_3, z_3) = (2,3,4)\).

Step 3: Compute the cross product of vectors PQ and PR

First, calculate vectors \( \vec{PQ} = Q - P = (0-1, 1-0, 0-1) = (-1, 1, -1) \) and \( \vec{PR} = R - P = (2-1, 3-0, 4-1) = (1, 3, 3) \).Now compute the cross product \(\vec{PQ} \times \vec{PR}\):\[\vec{PQ} \times \vec{PR} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \-1 & 1 & -1 \1 & 3 & 3 \ \end{vmatrix}\]\[= \mathbf{i}(1\cdot3 - (-1)\cdot3) - \mathbf{j}(-1\cdot3 - (-1)\cdot1) + \mathbf{k}(-1\cdot3 - 1\cdot1)\]\[= \mathbf{i}(3 + 3) - \mathbf{j}(-3 + 1) + \mathbf{k}(-3 - 1)\]\[= 6\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}\].

Step 4: Compute the magnitude of the cross product

The magnitude of the cross product \(\vec{PQ} \times \vec{PR}\) is given by: \[|\vec{PQ} \times \vec{PR}| = \sqrt{6^2 + 2^2 + (-4)^2} = \sqrt{36 + 4 + 16} = \sqrt{56} = 2\sqrt{14}\].

Step 5: Calculate the area of triangle PQR

The area of the triangle is half the magnitude of the cross product:\[\text{Area} = \frac{1}{2} \times |\vec{PQ} \times \vec{PR}| = \frac{1}{2} \times 2\sqrt{14} = \sqrt{14}.\]

Key Concepts

Understanding Cross Product

In 3D Geometry, the cross product is an essential vector operation, particularly when dealing with angles and areas. The cross product of two vectors results in a third vector that is perpendicular to the plane created by the original vectors. Mathematically, if you have two vectors \( \vec{A} = (a_1, a_2, a_3) \) and \( \vec{B} = (b_1, b_2, b_3) \), their cross product is a vector given by:

• \( \vec{A} \times \vec{B} = ((a_2b_3 - a_3b_2)\mathbf{i}, (a_3b_1 - a_1b_3)\mathbf{j}, (a_1b_2 - a_2b_1)\mathbf{k}) \)

This operation helps in finding a vector perpendicular to both \( \vec{A} \) and \( \vec{B} \). You can use the cross product to determine the plane of two vectors or calculating the area of shapes like triangles. Understand that the cross product is only applicable in three-dimensional space, providing a vector as a result. Keep in mind the right-hand rule which states that the direction of the cross product vector is determined by the orientation of your hand, aligning your fingers with the direction of the vectors.

Calculating the Area of a Triangle in 3D using Cross Product

One of the fascinating applications of the cross product is finding the area of a triangle in 3D space. When you have a triangle defined by vertices in a 3-dimensional coordinate system, the cross product can directly help you find the area. Let's consider a triangle with points \( P, Q, \) and \( R \). First, we calculate the vectors \( \vec{PQ} \) and \( \vec{PR} \) from the given points.

• Find the difference: \( \vec{PQ} = Q - P \)
• \( \vec{PR} = R - P \)

Once we have these vectors, we compute their cross product \( \vec{PQ} \times \vec{PR} \). The magnitude of this cross product vector gives twice the area of the triangle. It is because the cross product geometrically represents the parallelogram spanned by the two vectors. The area of the triangle is then half the magnitude of this cross product:

• \( \text{Area} = \frac{1}{2} |\vec{PQ} \times \vec{PR}| \)

Exploring Vector Operations in 3D

Vector operations in 3-dimensional geometry are central for modeling and solving many spatial problems. These operations include addition, subtraction, dot product (leading to scalar quantities), and the cross product (leading to vectors). For instance, when you calculate \( \vec{PQ} = (x_2-x_1, y_2-y_1, z_2-z_1) \), you perform a basic subtraction operation to find a vector direction from one point to another.The cross product, as discussed previously, is unique to 3D space and finds application in determining the normal vector to two other vectors.

• Vector addition: Sum components of vectors separately \( \vec{A} + \vec{B} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \)
• Vector subtraction: Difference of components \( \vec{A} - \vec{B} = (a_1 - b_1, a_2 - b_2, a_3 - b_3) \)

These operations not only define movement and direction in space but also calculate mutual relationships and interactions in physics and engineering problems. All these operations are crucial for comprehending the different approaches to problem-solving related to 3D geometry.