Q. 60 Prove property聽(3)... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 60 (page 628)

Prove property $(3)$

Short Answer

Expert verified

We had proved $u 脳 v = - (v 脳 u)$

Step by step solution

Step 1. Assumption Of Vectors

Let us assume that the two vectors $u, v$ be:

$u = a_{1} i + b_{1} j + c_{1} k and v = a_{2} i + b_{2} j + c_{2} k$

Step 2. Solve Left Hand Side

Now, we will solve the lest hand side of the equation,

$u 脳 v = |\begin{array}{l} i & j & k \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}|$

$= |\begin{array}{l} b_{1} & c_{1} \\ b_{2} & c_{2} \end{array}| i - |\begin{array}{l} a_{1} & c_{1} \\ a_{2} & c_{2} \end{array}| j + |\begin{array}{l} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array}| k$

$= (b_{1} c_{2} - b_{2} c_{1}) i - (a_{1} c_{2} - a_{2} c_{1}) j + (a_{1} b_{2} - a_{2} b_{1}) k$

$= - (b_{2} c_{1} - b_{1} c_{2}) i + (a_{2} c_{1} - a_{1} c_{2}) j - (a_{2} b_{1} - a_{1} b_{2}) k$

$= - [(b_{2} c_{1} - b_{1} c_{2}) i - (a_{2} c_{1} - a_{1} c_{2}) j + (a_{2} b_{1} - a_{1} b_{2}) k]$

Step 3. Find the Determinant Form

The above equation can be written in determinant form as,

$= - [\begin{array}{l} b_{2} & c_{2} \\ b_{1} & c_{1} \end{array} | i - | \begin{array}{l} a_{2} & c_{2} \\ a_{1} & c_{1} \end{array} | j + | \begin{matrix} a_{2} & b_{2} \\ a_{1} & b_{1} \end{matrix} 鈭� k]$

$= - |\begin{matrix} i & j & k \\ a_{2} & b_{2} & c_{2} \\ a_{1} & b_{1} & c_{1} \end{matrix}|$

$= - (v 脳 u)$

Hence Proved.

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91影视

Prove property $(3)$

Short Answer

Step by step solution

Step 1. Assumption Of Vectors

Step 2. Solve Left Hand Side

Step 3. Find the Determinant Form

One App. One Place for Learning.

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91影视

Prove property(3)

Short Answer

Step by step solution

Step 1. Assumption Of Vectors

Step 2. Solve Left Hand Side

Step 3. Find the Determinant Form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Prove property $(3)$