Question

Find the area of the following triangles \(T\). (The area of a triangle is half the area of the corresponding parallelogram.) The vertices of \(T\) are \(O(0,0,0), P(2,4,6),\) and \(Q(3,5,7)\)

Short answer

Answer: The area of the triangle T is \(\frac{\sqrt{72}}{2}\).

Step-by-step solution

Find the Vectors OP and OQ

First, let's find the vectors OP and OQ by calculating the component-wise difference between the coordinates of the points P and O, and between the points Q and O, respectively. $$ \begin{align*} \vec{OP} &= P - O = (2,4,6) - (0,0,0) = (2,4,6)\\ \vec{OQ} &= Q - O = (3,5,7) - (0,0,0) = (3,5,7) \end{align*} $$

Calculate the Vector Product of OP and OQ

The vector product can be calculated using the components of the vectors OP and OQ as follows: $$ \begin{align*} \vec{OP} \times \vec{OQ} &= \begin{pmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & 6 \\ 3 & 5 & 7 \end{pmatrix} \\ &= \hat{i}(4\cdot7 - 6\cdot5) - \hat{j}(2\cdot7 - 6\cdot3) + \hat{k}(2\cdot5 - 4\cdot3) \\ &= 2\hat{i} -8\hat{j} + 2\hat{k} \end{align*} $$

Find the Magnitude of the Vector Product

In order to find the area of the parallelogram formed by the vectors OP and OQ, we need to calculate the magnitude of their vector product: $$ \begin{align*} |\vec{OP} \times \vec{OQ}| &= \sqrt{(-2)^2 + (-8)^2 + 2^2}\\ &= \sqrt{4 + 64 +4}\\ &= \sqrt{72} \end{align*} $$ The magnitude of the vector product is \(\sqrt{72}\), representing the area of the parallelogram.

Find the Area of the Triangle

Since the area of the triangle is half the area of the parallelogram, we can now find the area of the triangle: $$ \text{Area}(T) = \frac{1}{2} \text{Area}(\text{Parallelogram}) = \frac{1}{2} \times \sqrt{72} = \frac{\sqrt{72}}{2} $$ The area of the triangle T is \(\frac{\sqrt{72}}{2}\).

Key concepts

Vector Product

The vector product, also known as the cross product, is a fundamental concept in vector calculus. It involves two vectors and results in another vector that is perpendicular to the plane formed by the initial vectors. This product is informative because its magnitude can be used to determine areas and volumes in three-dimensional space.
To compute the vector product \(\vec{A} \times \vec{B}\), you align the components of vectors \(\vec{A} = (a_1, a_2, a_3)\) and \(\vec{B} = (b_1, b_2, b_3)\) in a matrix, and use the determinant to find a vector perpendicular to both. This matrix is sometimes shown with the unit vectors \(\hat{i}, \hat{j}, \hat{k}\) like this:
\[\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} = \hat{i}(a_2b_3 - a_3b_2) - \hat{j}(a_1b_3 - a_3b_1) + \hat{k}(a_1b_2 - a_2b_1)\]

Magnitude of Vector

The magnitude of a vector, often called its length or norm, is a measure of how long the vector is. For a vector \(\vec{V} = (v_1, v_2, v_3)\), the magnitude is computed using the formula:
\[|\vec{V}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\]
This formula can be applied to any dimensional space, not just three-dimensional spaces. Magnitude is crucial when determining the size of vector projections and when calculating physical quantities such as velocity and force. It provides an easy way to quantify vectors' scale or size.
In exercises involving the vector product, the magnitude is particularly useful in finding the area of geometric shapes like parallelograms. The magnitude of the vector product of two vectors gives the area of the parallelogram formed by those vectors.

Area of Triangle

Finding the area of a triangle using vectors is a powerful application in vector calculus. Interestingly, you can use the vector product to determine this area indirectly. The steps involve finding the vector product first, then using its magnitude.
The formula for the area of a triangle when vectors are involved is:
\[\text{Area of Triangle} = \frac{1}{2} |\vec{A} \times \vec{B}|\]
Here, \(\vec{A}\) and \(\vec{B}\) represent the vectors forming two sides of the triangle. Initially, you calculate their vector product, which gives the area of the parallelogram created by these vectors. By halving this area, you derive the area of the triangle, since a triangle is just half of a parallelogram shaped by the same vectors.

Parallelogram

In geometry, a parallelogram is a four-sided shape with opposite sides that are parallel and equal in length. Parallelograms hold certain symmetrical properties and serve as a foundation for calculating other geometric areas, such as those of triangles.
The area of a parallelogram can be found using vectors by taking the cross product of the vectors that define its sides and then finding the magnitude of the resulting vector:
\[\text{Area of Parallelogram} = |\vec{A} \times \vec{B}|\]
This method uses the principle that the magnitude of the cross product of two vectors gives the area of the parallelogram they span. This capability is critical in physics and engineering because it provides a concise way to calculate two-dimensional areas within three-dimensional spaces.