Q4E Consider the area of the triangl... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q4E (page 305)

Consider the area of the triangle with vertices $[\begin{matrix} a_{1} \\ a_{2} \end{matrix}], [\begin{matrix} b_{1} \\ b_{2} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \end{matrix}]$ . ExpressAin terms of $[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ 1 & 1 & 1 \end{matrix}]$

Short Answer

Expert verified

Therefore, theArea of triangle express in terms of $\det [\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ 1 & 1 & 1 \end{matrix}]$ is,

role="math" localid="1660717647418" $\frac{1}{2} |\det [\begin{matrix} b_{1} - a_{1} & c_{1} - a_{1} \\ b_{2} - a_{2} & c_{2} - a_{2} \end{matrix}]| = \frac{1}{2} |\det [\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ 1 & 1 & 1 \end{matrix}]|$

Step by step solution

Definition.

Basically, it is equal to half of the Area of the parallelogram.

The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle.

$A = 1 / 2 脳 b 脳 h$

Given.

Given Matrix,

$[\begin{matrix} a_{1} \\ a_{2} \end{matrix}],$

And

$[\begin{matrix} b_{1} \\ b_{2} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \end{matrix}]$

To find the area of the triangle express in terms of det [a1b1c1a2b2c2111]

The area of this triangle,whose vertices are,

$[\begin{matrix} a_{1} \\ a_{2} \end{matrix}], [\begin{matrix} b_{1} \\ b_{2} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \end{matrix}]$

Is the same as the area of a triangle formed by,

$[\begin{matrix} b_{1} - a_{1} \\ b_{2} - a_{2} \end{matrix}]$

And

$[\begin{matrix} c_{1} - a_{1} \\ c_{2} - a_{2} \end{matrix}]$

$\frac{1}{2} |\det [\begin{matrix} b_{1} - a_{1} & c_{1} - a_{1} \\ b_{2} - a_{2} & c_{2} - a_{2} \end{matrix}]| = \frac{1}{2} |(b_{1} - a_{1}) (c_{2} - a_{2}) - (b_{2} - a_{2}) (c_{1} - a_{1})| \frac{1}{2} |\det [\begin{matrix} b_{1} - a_{1} & c_{1} - a_{1} \\ b_{2} - a_{2} & c_{2} - a_{2} \end{matrix}]| = \frac{1}{2} |b_{1} a_{2} - b_{1} a_{2} - a_{1} c_{2} - b_{2} c_{1} + b_{2} c_{1} + b_{2} a_{1} + a_{2} c_{1} + a_{1} a_{2}| \frac{1}{2} |\det [\begin{matrix} b_{1} - a_{1} & c_{1} - a_{1} \\ b_{2} - a_{2} & c_{2} - a_{2} \end{matrix}]| = \frac{1}{2} |\det [\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ 1 & 1 & 1 \end{matrix}]|$

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91影视

Consider the area of the triangle with vertices $[\begin{matrix} a_{1} \\ a_{2} \end{matrix}], [\begin{matrix} b_{1} \\ b_{2} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \end{matrix}]$ . ExpressAin terms of $[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ 1 & 1 & 1 \end{matrix}]$

Short Answer

Step by step solution

Definition.

Given.

To find the area of the triangle express in terms of det [a1b1c1a2b2c2111]

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91影视

Consider the area of the triangle with vertices [a1a2],[b1b2],[c1c2]. ExpressAin terms of[a1b1c1a2b2c2111]

Short Answer

Step by step solution

Definition.

Given.

To find the area of the triangle express in terms of det [a1b1c1a2b2c2111]

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.