Q. 78 Let u=(u1,u2,u3),聽v=(v1,v2,v3),... [FREE SOLUTION]

Chapter 10: Q. 78 (page 826)

Let $u = (u_{1}, u_{2}, u_{3}), v = (v_{1}, v_{2}, v_{3}), and w = (w_{1}, w_{2}, w_{3})$ . Show that
$u 路 (v 脳 w) = \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}]$

Short Answer

Expert verified

Hence, we prove that $u 路 (v 脳 w) = \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}]$

Step by step solution

Step 1. Given Information

Let $u = (u_{1}, u_{2}, u_{3}), v = (v_{1}, v_{2}, v_{3}), and w = (w_{1}, w_{2}, w_{3})$ . Show that

role="math" localid="1650039947035" $u 路 (v 脳 w) = \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}]$

Step 2. Given that u=(u1,u2,u3), v=(v1,v2,v3), and w=(w1,w2,w3)

We firstly finding the value of $v 脳 w$

role="math" localid="1650039221614"

v 脳 w = \det [\begin{matrix} i & j & k \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}] v 脳 w = i |\begin{matrix} v_{2} & v_{3} \\ w_{2} & w_{3} \end{matrix}| - j |\begin{matrix} v_{1} & v_{3} \\ w_{1} & w_{3} \end{matrix}| + k |\begin{matrix} v_{1} & v_{2} \\ w_{1} & w_{2} \end{matrix}| v 脳 w = i (v_{2} w_{3} - v_{3} w_{2}) - j (v_{1} w_{3} - v_{3} w_{1}) + k (v_{1} w_{2} - v_{2} w_{1}) v 脳 w = (v_{2} w_{3} - v_{3} w_{2}, v_{1} w_{3} - v_{3} w_{1}, v_{1} w_{2} - v_{2} w_{1})

Step 3. Now finding the value of u·(v×w)

$u 路 (v 脳 w) = (u_{1}, u_{2}, u_{3}) 路 (v_{2} w_{3} - v_{3} w_{2}, v_{1} w_{3} - v_{3} w_{1}, v_{1} w_{2} - v_{2} w_{1}) u 路 (v 脳 w) = u_{1} (v_{2} w_{3} - v_{3} w_{2}) + u_{2} (v_{1} w_{3} - v_{3} w_{1}) + u_{3} (v_{1} w_{2} - v_{2} w_{1}) u 路 (v 脳 w) = u_{1} v_{2} w_{3} - u_{1} v_{3} w_{2} + u_{2} v_{1} w_{3} - u_{2} v_{3} w_{1} + u_{3} v_{1} w_{2} - u_{3} v_{2} w_{1}$

Step 4. Now solving the detu1u2u3v1v2v3w1w2w3

$\det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}] = u_{1} |\begin{matrix} v_{2} & v_{3} \\ w_{2} & w_{3} \end{matrix}| - u_{2} |\begin{matrix} v_{1} & v_{3} \\ w_{1} & w_{3} \end{matrix}| + u_{3} |\begin{matrix} v_{1} & v_{2} \\ w_{1} & w_{2} \end{matrix}| \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}] = u_{1} (v_{2} w_{3} - v_{3} w_{2}) - u_{2} (v_{1} w_{3} - v_{3} w_{1}) + u_{3} (v_{1} w_{2} - v_{2} w_{1}) \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}] = u_{1} v_{2} w_{3} - u_{1} v_{3} w_{2} - u_{2} v_{1} w_{3} - u_{2} v_{3} w_{1} + u_{3} v_{1} w_{2} - u_{3} v_{2} w_{1}$

Hence, $u 路 (v 脳 w) = \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}]$

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91影视

Let $u = (u_{1}, u_{2}, u_{3}), v = (v_{1}, v_{2}, v_{3}), and w = (w_{1}, w_{2}, w_{3})$ . Show that
$u 路 (v 脳 w) = \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}]$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Given that u=(u1,u2,u3), v=(v1,v2,v3), and w=(w1,w2,w3)

Step 3. Now finding the value of u·(v×w)

Step 4. Now solving the detu1u2u3v1v2v3w1w2w3

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91影视

Let u=(u1,u2,u3),v=(v1,v2,v3),andw=(w1,w2,w3). Show thatu路(v脳w)=detu1u2u3v1v2v3w1w2w3

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Given that u=(u1,u2,u3), v=(v1,v2,v3), and w=(w1,w2,w3)

Step 3. Now finding the value of u·(v×w)

Step 4. Now solving the detu1u2u3v1v2v3w1w2w3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Probability and Statistics

Geometry

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let $u = (u_{1}, u_{2}, u_{3}), v = (v_{1}, v_{2}, v_{3}), and w = (w_{1}, w_{2}, w_{3})$ . Show that
$u 路 (v 脳 w) = \det [\begin{matrix} u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix}]$