Problem 15 - Calculus: Early Transcendentals

Question

Any three points \(P_{1}\left(x_{1}, y_{1}, z_{1}\right), P_{2}\left(x_{2}, y_{2}, z_{2}\right), P_{3}\left(x_{3}, y_{3}, z_{3}\right),\) lie in a plane and form a triangle. The triangle inequality says that \(d\left(P_{1}, P_{3}\right) \leq d\left(P_{1}, P_{2}\right)+d\left(P_{2}, P_{3}\right) .\) Prove the triangle inequality using either algebra (messy) or the law of cosines (less messy).

Step-by-Step Solution

Answer

Use the law of cosines to show \(c \leq a + b\).

Step 1: Recognize the Law of Cosines Formula

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides of length \(a\), \(b\), and \(c\), opposite the angles \(A\), \(B\), and \(C\) respectively, the law of cosines states that:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Step 2: Express Distance Using the Distance Formula

The distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3-dimensional space is calculated using:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Step 3: Apply Law of Cosines to Triangle Formed by Points

For points \(P_1, P_2, P_3\), let the distances between these points represent the sides of a triangle:- \(a = d(P_1, P_2)\)- \(b = d(P_2, P_3)\)- \(c = d(P_1, P_3)\)Using the law of cosines:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where \(C\) is the angle at \(P_2\).

Step 4: Simplify Using the Non-negative Property of Cosine

Since the range of \(\cos(C)\) is from \(-1\) to \(1\), we have the minimum value of \(c^2\) when \(\cos(C) = 1\). Thus, it simplifies to:\[ c^2 \leq a^2 + b^2 \] Consequently:\[ c \leq \sqrt{a^2 + b^2} = a + b \]

Step 5: Conclusion: Verifying the Triangle Inequality

Hence, we have demonstrated using the law of cosines that:\[ d(P_1, P_3) \leq d(P_1, P_2) + d(P_2, P_3) \] This confirms that the triangle inequality holds for the points \(P_1, P_2, P_3\).

Key Concepts

Law of Cosines

The Law of Cosines is like a bridge connecting the angles and the side lengths in a triangle. It is particularly useful when you have a triangle and you know two sides and the angle between them, or when you know all the sides but need to find an angle. This concept allows us to calculate an unknown side of the triangle if two sides and their included angle are known.

The formula for the Law of Cosines is given by:

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

Here, \(c\) is the side opposite angle \(C\), and \(a\) and \(b\) are the other two sides of the triangle. The Law of Cosines can be viewed as a generalization of the Pythagorean theorem, which applies even if your triangle isn’t a right triangle.

In situations where you're dealing with an isosceles triangle, or a triangle with certain symmetry, the Law of Cosines can simplify problem-solving, turning complex geometry into manageable algebra.

Distance Formula in 3D

The Distance Formula in 3D allows us to calculate the length of a segment between two points in three-dimensional space. Understanding this formula is vital when dealing with problems that involve points in 3D.

The formula is expressed as:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)

This accounts for the change in each of the spatial dimensions—x, y, and z. Conceptually, this formula is an extension of the Pythagorean theorem to three dimensions, where each squared term represents the squared distance component along each respective axis.

It is crucial when determining the distances between points, as seen in physics, engineering, and various geometric calculations. Using this formula ensures precision when measuring and calculating distances wisely.

Non-negative Property of Cosine

The cosine of an angle is a critical trigonometric function that has useful properties. One of these is its non-negative behavior when used under certain conditions in the realm of geometry. The cosine function has a range of values between (-1) and 1. This means for angles between 0° and 90°, the cosine is always positive and decreases as the angle approaches 90°.

This leads to a vital implication in terms of analysing triangles. In the context of the Law of Cosines, knowing that \(\cos(C)\) can't go below -1 allows us to make meaningful statements about the side lengths:

• If \(\cos(C) = 1\), the Law of Cosines simplifies beautifully to the Triangle Inequality, showing straightforwardly that \(c \leq a + b\).

Positive cosine values reinforce certain triangle properties such as the triangle inequality, ensuring that the geometry of triangles behaves predictably, which is paramount in geometric proofs and problem-solving.