Problem 46 - Precalculus with Limits

Question

In Exercises 45-48, use vectors to find the interior angles of the triangle with the given vertices. \((-3, -4)\), \((1, 7)\), \((8, 2)\)

Step-by-Step Solution

Answer

The interior angles of the triangle having vertices (-3, -4), (1, 7) and (8, 2) can be calculated using the above steps. Once the angles are computed, verify that they add up to 180 degrees to validate the solution.

Step 1: Define the Vectors

The first step is to understand the triangle's vertices given and define the corresponding vector for each side. Here we have vertices at (-3, -4), (1, 7), (8, 2). Hence the vectors for AB, BC and CA can be defined as \(\vec{AB} = (1 - (-3), 7 - (-4))\), \(\vec{BC} = (8 - 1, 2 - 7)\) and \(\vec{CA} = (-3 - 8, -4 - 2)\) respectively.

Step 2: Compute Dot Products

Compute the Dot product of the vectors which will be used in the next step to calculate the respective angles. The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) can be found using the formula, \(\vec{A} \cdot \vec{B} = ||A|| ||B|| cos(Θ)\) where \(||A||\) and \(||B||\) are the magnitudes of \(\vec{A}\) and \(\vec{B}\) respectively. Therefore, the dot products of the vectors \(AB . BC\), \(BC . CA\) and \(CA . AB\) will be required next.

Step 3: Calculate the Angles

The cosine of the angle between two vectors can be calculated using the formula \(\cos(Θ) = \frac{\vec{A} . \vec{B}}{||A|| ||B||}\). Hence, if \(Θ_{A}\), \(Θ_{B}\) and \(Θ_{C}\) are the angles at vertices A, B and C respectively, they can be calculated by rearranging the above formula to obtain the angles from the respective cosines.

Step 4: Convert to Degrees

Since the above formula gives the result in radians, multiply it by \( \frac{180}{\pi} \) to convert the angles to degrees.

Key Concepts

Dot Product

The dot product is a crucial operation in vector mathematics. It helps in finding the angle between two vectors, which is essential for solving problems involving vector geometry. The dot product of two vectors \( \vec{A} = (a_1, a_2) \) and \( \vec{B} = (b_1, b_2) \) is calculated as:

• \( \vec{A} \cdot \vec{B} = a_1 \times b_1 + a_2 \times b_2 \)

This result is a scalar, which means it’s a single number, not a vector. The significance of the dot product becomes clear in its geometric interpretation. If the dot product is zero, the vectors are perpendicular. Moreover, by rearranging the dot product formula, the angle \( \Theta \) between them can be found.

Interior Angles

Interior angles of a triangle that are formed by the triangle's vertices can be determined using vectors. Each angle depends on the dot product between the two vectors that form the sides of the angle. To find an angle \( \Theta \) at vertex B for instance, you:

• Calculate the vectors for the sides joining at B, such as \( \vec{AB} \) and \( \vec{BC} \).
• Use the relationship \( \cos(\Theta) = \frac{\vec{AB} \cdot \vec{BC}}{||\vec{AB}|| ||\vec{BC}||} \).

This cosine value can then be used to find the angle \( \Theta \) by taking its inverse cosine. This method allows you to precisely calculate each interior angle in the triangle.

Triangle Vertices

The vertices of a triangle are the points where two sides meet. In coordinate geometry, these vertices are represented with coordinates, for example, \( (-3, -4) \), \( (1, 7) \), and \( (8, 2) \). Defining each side of the triangle as a vector helps to analyze and solve various geometric problems.

• Vector \( \vec{AB} \) is derived from points A and B: \( (1 - (-3), 7 - (-4)) = (4, 11) \).
• Vector \( \vec{BC} \) from B to C is \( (8 - 1, 2 - 7) = (7, -5) \).
• Vector \( \vec{CA} \) from C to A is \( (-3 - 8, -4 - 2) = (-11, -6) \).

These vectors are the building blocks needed to compute other triangle properties like the angles.

Magnitude of Vectors

The magnitude of a vector is a measure of its length. It is found using the formula derived from the Pythagorean Theorem. For a vector \( \vec{A} = (x, y) \), its magnitude \( ||\vec{A}|| \) is:

• \( ||\vec{A}|| = \sqrt{x^2 + y^2} \)

Magnitude is essential for calculating the angle between vectors. For a triangle's interior angles, finding the magnitude of each vector, such as \( ||\vec{AB}|| \), \( ||\vec{BC}|| \), and \( ||\vec{CA}|| \), helps establish the relationship needed to find angle sizes.The magnitudes also help determine the relative size of each side of the triangle.