Textbook Question
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i - 10j
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 46
In Exercises 45–46, find the area of the triangle with the given vertices. Round to the nearest square unit. (-2, -3), (-2, 2), (2, 1)
Verified step by step guidance1
Identify the coordinates of the vertices of the triangle: \(A(-2, -3)\), \(B(-2, 2)\), and \(C(2, 1)\).
Use the formula for the area of a triangle given coordinates \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\):
\(\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|\)
Substitute the coordinates into the formula:
\(\text{Area} = \frac{1}{2} \left| (-2)(2 - 1) + (-2)(1 - (-3)) + 2((-3) - 2) \right|\)
Simplify the expression inside the absolute value by performing the operations inside the parentheses and then the multiplications.
Calculate the absolute value of the simplified expression, multiply by \(\frac{1}{2}\), and then round the result to the nearest whole number to find the area of the triangle.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Coordinate Geometry and Triangles
Understanding how to represent points in the coordinate plane is essential. Each vertex of the triangle is given as an (x, y) coordinate, which allows the use of algebraic methods to calculate distances, slopes, and areas.
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Area of a Triangle Using Coordinates
The area of a triangle with vertices at coordinates can be found using the formula: Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. This formula uses the determinant method to calculate the absolute value of the signed area.
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Rounding and Approximation
After calculating the exact area, rounding to the nearest square unit is necessary. This involves understanding decimal values and applying standard rounding rules to present the final answer in a simplified, practical form.
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Related Practice
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v. v = i + j
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = i - j
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Textbook Question
In Exercises 43–44, use the given measurements to solve the following triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree. a = 400, b = 300
Textbook Question
In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 6, θ = 30°
Textbook Question
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i + 10j