Explanation Given: The points ( 1 , 3 , − 1 ) , ( 2 , − 1 , 3 ) and ( 4 , 1 , 1 ) form a triangle Key concepts applied: There are two methods one can use to find the area of a triangle given three vertices. One could find the length of one of the altitudes of the triangle and proceed to find the area using the traditional area of the triangle formula given by Area of triangle, A = 1 2 × b a s e × a l t i t u d e However, in three dimensions, this method could get tedious. Instead one could consider using the formula for the area of the parallelogram. The area of a triangle within a parallelogram is one-half the area of the parallelogram Thus, area of the triangle is given by Area of triangle, A = 1 2 Area of parallelogram = 1 2 ‖ a × b ‖ Where a and b are vectors representing two sides of the triangle/parallelogram. The vectors a and b can be calculated from point vectors denoting the vertices of the triangle/parallelogram. Vector joining two points If P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) are any two points, then the vector joining P 1 and P 2 is given by, P 1 P 2 → = O P 2 → − O P 1 → Vector cross product The cross product of two vectors a and b , when their magnitudes ‖ a ‖ and ‖ b ‖ and the angle between them θ are known is given by the formula, a × b = ‖ a ‖ ⋅ ‖ b ‖ ⋅ sin ( θ ) n Where n is the unit vector at right angles to both a and b The cross product of the given two vectors a = 〈 a 1 , a 2 , a 3 〉 and b = 〈 b 1 , b 2 , b 3 〉 can also be calculated using the formula below, a × b = 〈 c 1 , − c 2 , c 3 〉 where c 1 = a 2 b 3 − a 3 b 2 c 2 = a 1 b 3 − a 3 b 1 c 3 = a 1 b 2 − a 2 b 1 Considering i, j and k to be the standard unit vectors in R 3 (assuming the right handed coordinate system), we can write the cross product in the form of the determinant as follows, a × b = | i j k a i a j a k b i b j b k | The cross product of two vectors is another vector that is at right angles to both. Evaluating determinants The determinant of a square matrix A = [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] is calculated as follows, | A | = a 1 ( b 2 c 3 − b 3 c 2 ) − a 2 ( b 1 c 3 − b 3 c 1 ) + a 3 ( b 1 c 2 − b 2 c 1 ) Magnitude of a vector The magnitude of a vector in Cartesian coordinates is the square root of the sum of the squares of its coordinates. For a three-dimensional vector, the length of the vector is found using the following formula. ‖ a ‖ = a x 2 + a y 2 + a z 2 Formula used: Area of a triangle Area of triangle = 1 2 Area of parallelogram = 1 2 ‖ a × b ‖ Where a and b are vectors representing two sides of the triangle/parallelogram...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Videos

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 13, Problem 37CRE

To determine

To calculate:

The area of the given triangle using cross product

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 36CRE

Chapter 13, Problem 38CRE

Students have asked these similar questions

For each given function f(x) find f'(x) using the rules learned in section 9.5. 1. f(x)=x32 32x 2. f(x)=7x+13 3. f(x) = x4 4. f(x) = √√x³ 5. f(x) = 3x²+ 3 x2

Find: lim x →-6 f (x) limx-4 f (x) lim x-1 f (x) lim x →4 f (x) (-6,3) • (-1,5) -8 -7 (-6,-2) 4+ (4,5) (4,2) • (-1,1) -6

3 2 Find: ƒ(1) lim f(x) 14-x 2 ƒ(2) lim f(x) x-2- lim f(x) x+2+ lim f(x) x→4 3 y=f(x)

Chapter 13 Solutions

Calculus - Standalone book

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The area of the given triangle using cross product

Solution Summary: The author explains how to calculate the area of a given triangle using cross product.

Concept explainers

Videos

To calculate: The area of the given triangle using cross product

Want to see the full answer?

Chapter 13 Solutions

To calculate:

The area of the given triangle using cross product