Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

If u and v are nonzero vectors in $ℝ^{3}$ , why do the equations role="math" localid="1649263352081" $u \cdot (u \times v) = 0$ and $v \cdot (u \times v) = 0$ tell us that the cross product is orthogonal to both u and v?

We prove that $u \cdot (u \times v) = 0 and v \cdot (u \times v) = 0$ tell us that the cross product is orthogonal to both u and v.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

If u and v are nonzero vectors in $ℝ^{3}$ , why do the equations $u \cdot (u \times v) = 0$ and $v \cdot (u \times v) = 0$ tell us that the cross product is orthogonal to both u and v?

Step 2. u and v are nonzero vectors in ℝ3.

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

The determinant of a 3 × 3 matrix is

$u \times v = \det [\begin{matrix} i & j & k \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{matrix}]$

Step 3. Then, by the definition of the cross product,

$u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$

Now proving the equation $u \cdot (u \times v) = 0$

$u \cdot (u \times v) = (u_{1}, u_{2}, u_{3}) \cdot (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1}) u \cdot (u \times v) = u_{1} (u_{2} v_{3} - u_{3} v_{2}) + u_{2} (u_{3} v_{1} - u_{1} v_{3}) + u_{3} (u_{1} v_{2} - u_{2} v_{1}) u \cdot (u \times v) = 0$

Step 4. Now proving the equationv·(u×v)=0.

$v \cdot (u \times v) = (v_{1}, v_{2}, v_{3}) \cdot (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1}) v \cdot (u \times v) = v_{1} (u_{2} v_{3} - u_{3} v_{2}) + v_{2} (u_{3} v_{1} - u_{1} v_{3}) + v_{3} (u_{1} v_{2} - u_{2} v_{1}) v \cdot (u \times v) = 0$

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Calculus

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Short Answer

If u and v are nonzero vectors in $ℝ^{3}$ , why do the equations role="math" localid="1649263352081" $u \cdot (u \times v) = 0$ and $v \cdot (u \times v) = 0$ tell us that the cross product is orthogonal to both u and v?

Step by Step Solution

Step 1. Given Information

Step 2. u and v are nonzero vectors in ℝ3.

Step 3. Then, by the definition of the cross product,

Step 4. Now proving the equationv·(u×v)=0.

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Calculus

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Short Answer

If u and v are nonzero vectors in ℝ3, why do the equations role="math" localid="1649263352081" u·(u×v)=0 and v·(u×v)=0 tell us that the cross product is orthogonal to both u and v?

Step by Step Solution

Step 1. Given Information

Step 2. u and v are nonzero vectors in ℝ3.

Step 3. Then, by the definition of the cross product,

Step 4. Now proving the equationv·(u×v)=0.

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Pure Maths

If u and v are nonzero vectors in $ℝ^{3}$ , why do the equations role="math" localid="1649263352081" $u \cdot (u \times v) = 0$ and $v \cdot (u \times v) = 0$ tell us that the cross product is orthogonal to both u and v?