Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use vectors to find the interior angles of the triangle with the given vertices. $$(-3,5), (-1,9), (7,9)$$

Step-by-Step Solution

Answer

The measure of the interior angle of the triangle at vertex A, when calculated using vector methods, results in \( \theta = \arccos \left( \frac{36} {\sqrt{20} \sqrt{116}} \right) \)

Step 1: Construct the Vectors

The given points are (-3,5), (-1,9), (7,9). Let's assign these points as A, B and C respectively. We can form two vectors AB and AC from these points. To get vector AB, subtract the coordinates of A from B, and to get AC, subtract the coordinates of A from C. Vector AB = B - A = (-1 - (-3), 9 - 5)= (2,4), Vector AC= C - A = (7 - (-3), 9 - 5) = (10,4)

Step 2: Calculate the Dot Product

Now, calculate the dot product of the vectors AB and AC. The dot product of two vectors (a, b) and (c, d) is given by \( ac + bd \). The dot product of AB and AC is, therefore, (2 * 10) + (4 * 4) = 36

Step 3: Calculate the Magnitude of the Vectors

The magnitude of a vector (a, b) is given by \( \sqrt{{a^2} + {b^2}} \). Thus, the magnitude of AB is \( \sqrt{{2^2} + {4^2}} = \sqrt{20} \), and the magnitude of AC is \( \sqrt{{10^2} + {4^2}} =\sqrt{116} \)

Step 4: Use the Cosine Rule to Find the Angle

The angle \( \theta \) between two vectors can be found using the formula \( \cos \theta = \frac{{AB \cdot AC}}{{|AB| |AC|}} \) where the dot (.) represents the dot product and |AB|, |AC| represent the magnitude of AB and AC respectively. Hence \( \cos \theta = \frac{{36}}{{\sqrt{20} \sqrt{116}}} \) . Therefore, \( \theta = \arccos \left( \frac{36} {\sqrt{20} \sqrt{116}} \right) \)

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation for vector calculations in geometry. It measures the extent to which two vectors point in the same direction. If you have two vectors, say \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), the dot product is computed as \( a_1 b_1 + a_2 b_2 \). This results in a scalar.

• If the dot product is positive, the vectors point somewhat in the same direction.
• If it is zero, the vectors are perpendicular.
• A negative result indicates that the vectors are pointing in opposite directions.

In our triangle exercise, the dot product helps find the angle between vectors AB and AC by determining how aligned these vectors are.

Magnitude of Vectors

Magnitude gives the length or size of a vector, similar to finding the length of a line segment connected by two points. For a vector \( \mathbf{v} = (v_1, v_2) \), its magnitude is calculated using the formula \( \sqrt{v_1^2 + v_2^2} \). This expression is derived from the Pythagorean theorem.

• The magnitude is always a non-negative number.
• It represents the Euclidean distance from the origin to the point \( (v_1, v_2) \) on a plane.

In calculating the angle between two vectors like AB and AC, knowing their magnitudes is crucial as it scales the vectors correctly for further computations. The cleaner the calculation of magnitudes, the more accurate the angle determination.

Cosine Rule

The cosine rule in vectors is essential for finding the angle between two vectors. If \( \theta \) is the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \), it uses \[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \] This formula gives the cosine of the angle using the dot product and the magnitudes of the vectors.

• Divide the dot product by the product of the vectors' magnitudes.
• Withdraw the arccos function to find the angle \( \theta \) from cosine value.

This rule is a part of trigonometry applied in vectors to compute interior angles in a triangle, such as the one formed by points A, B, and C in our example.

Interior Angles of a Triangle

Finding the interior angles of a triangle is made efficient using vectors and the tools we discussed. Once vectors are formed using points of a triangle, the dot product and magnitudes provide a path to uncover the angles. A triangle always has angles that sum to 180 degrees.

• First, form vectors based on the triangle's vertices.
• Use the cosine rule to compute each angle using dot product and magnitudes.
• Accompany this with vector methods to ensure comprehensive angle measures.

Understanding each angle within the triangle helps solve various problems in geometry and physics, applying effective principles like those using vectors efficiently.