Problem 34 - Calculus for Biology and Medicine

Question

A triangle has vertices at coordinates \(P=(2,1,5), Q=\) \((-1,3,7)\), and \(R=(2,-4,1)\) (a) Compute the lengths of all three sides. (b) Compute all three angles in both radians and degrees.

Step-by-Step Solution

Answer

Side lengths: \(PQ = \sqrt{17}, QR = \sqrt{94}, RP = \sqrt{41}\). Angles can be found using the cosine law.

Step 1: Understanding the distance formula

To calculate the length of a side of a triangle in 3D, we use the distance formula: \[\text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]This formula will be applied to each pair of vertices for triangle sides PQ, QR, and RP.

Step 2: Calculating length of side PQ

Using the coordinates of points P and Q: \[PQ = \sqrt{((-1) - 2)^2 + (3 - 1)^2 + (7 - 5)^2}\]\[= \sqrt{(-3)^2 + 2^2 + 2^2}\]\[= \sqrt{9 + 4 + 4}\]\[= \sqrt{17}\]

Step 3: Calculating length of side QR

Using the coordinates of points Q and R: \[QR = \sqrt{(2 - (-1))^2 + (-4 - 3)^2 + (1 - 7)^2}\]\[= \sqrt{3^2 + (-7)^2 + (-6)^2}\]\[= \sqrt{9 + 49 + 36}\]\[= \sqrt{94}\]

Step 4: Calculating length of side RP

Using the coordinates of points R and P: \[RP = \sqrt{(2 - 2)^2 + (1 - (-4))^2 + (5 - 1)^2}\]\[= \sqrt{0 + 5^2 + 4^2}\]\[= \sqrt{0 + 25 + 16}\]\[= \sqrt{41}\]

Step 5: Finding Angles Using the Cosine Law

The cosine law relates the sides of a triangle to one of its angles: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]Rearranging, we find \(C\): \[\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\]This formula will help us calculate angles using the side lengths.

Step 6: Calculating Angle PQR

Use the lengths: - \( PQ = \sqrt{17} \)- \( QR = \sqrt{94} \)- \( RP = \sqrt{41} \)Using the cosine law for angle \( \theta_{PQR} \):\[\cos(\theta_{PQR}) = \frac{PQ^2 + QR^2 - RP^2}{2 \times PQ \times QR}\]\[= \frac{17 + 94 - 41}{2 \times \sqrt{17} \times \sqrt{94}}\]\[= \frac{70}{2 \times \sqrt{17 \times 94}}\]Calculate the angle using inverse cosine to find \(\theta_{PQR}\) in radians and then convert to degrees.

Step 7: Calculating Angle QRP

For angle \( \theta_{QRP} \),\[\cos(\theta_{QRP}) = \frac{QR^2 + RP^2 - PQ^2}{2 \times QR \times RP}\]\[= \frac{94 + 41 - 17}{2 \times \sqrt{94} \times \sqrt{41}}\]\[= \frac{118}{2 \times \sqrt{94 \times 41}}\]Calculate angle \(\theta_{QRP}\) using the inverse cosine in radians and convert it to degrees.

Step 8: Calculating Angle RQP

For angle \( \theta_{RQP} \),\[\cos(\theta_{RQP}) = \frac{RP^2 + PQ^2 - QR^2}{2 \times RP \times PQ}\]\[= \frac{41 + 17 - 94}{2 \times \sqrt{41} \times \sqrt{17}}\]\[= \frac{-36}{2 \times \sqrt{41 \times 17}}\]Evaluate and determine the angle in radians first and then convert it to degrees.

Key Concepts

Distance Formula

3D Geometry often involves calculating distances between points in space. This can be visualized by imagining how far apart various points are if they were connected by a straight line. In our exercise, we dealt with vertices P, Q, and R in a 3D space. The distance formula helps us compute the length of these lines.

The formula used is:

• \[ \text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \]

Here's what each term means:

• \((x_2, y_2, z_2)\) and \((x_1, y_1, z_1)\) are the coordinates of the two points.
• Subtracting the same coordinate values (e.g., \(x_2 - x_1\)) helps find the change in each dimension.
• The squares of these differences are summed and then square-rooted to find the total distance.

This calculation method maintains accuracy in multi-dimensional analysis, crucial in handling problems involving three-dimensional spaces, like calculating side lengths for triangles in 3D.

Cosine Law

The Cosine Law is an extension of the Pythagorean theorem but adapted for all triangles, not just right triangles. It's especially useful in 3D geometry where many triangles do not have right angles. In the exercise, we used it to find the angles of the triangle given the side lengths.

The Cosine Law formula is:

• \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Here's a simple breakdown:

• \(c\) is the length of the side opposite the angle you're calculating (in this case, \(C\)).
• \(a\) and \(b\) are the lengths of the other two sides.
• Rearrange to find \(\cos(C)\): \[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]

The Cosine Law allows you to solve for one of the triangle's angles when you know the lengths of all three sides. It's a powerful tool in 3D geometry, ensuring we understand the spatial relation between sides and angles.

Triangle Angles

Understanding triangle angles is key in geometry. Regardless of the context, a triangle's internal angles always sum up to \(180^\circ\) or \(\pi\) radians. The exercise involved finding angles in both degrees and radians, which is a common practice since both units are often used interchangeably.

In the problem, we calculated the angles using the cosine law, and then converted our results:

• To convert an angle from radians to degrees, you use: \[ \text{Degrees} = \, \text{Radians} \times \frac{180}{\pi} \]
• Conversely, to go from degrees to radians: \[ \text{Radians} = \, \text{Degrees} \times \frac{\pi}{180} \]

Understanding the conversion is critical in both educational exercises and real-world applications, ensuring you can interpret angles correctly no matter the unit. This dual approach to angles ensures that you're equipped for varied mathematical and engineering problems.