Q. 7 If u and v are nonzero vectors i... [FREE SOLUTION]

Chapter 10: Q. 7 (page 823)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

If u and v are nonzero vectors in ${鈩�}^{3}$ , why do the equations $u 路 (u 脳 v) = 0$ and $v 路 (u 脳 v) = 0$ tell us that the cross product is orthogonal to both u andv?

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 脳 3 matrix is

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

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

$u 脳 v = (u_{2} v_{3} 鈭� u_{3} v_{2}, u_{3} v_{1} 鈭� u_{1} v_{3}, u_{1} v_{2} 鈭� u_{2} v_{1})$

Now proving the equation $u 路 (u 脳 v) = 0$

$u 路 (u 脳 v) = (u_{1}, u_{2}, u_{3}) 路 (u_{2} v_{3} 鈭� u_{3} v_{2}, u_{3} v_{1} 鈭� u_{1} v_{3}, u_{1} v_{2} 鈭� u_{2} v_{1}) u 路 (u 脳 v) = u_{1} (u_{2} v_{3} 鈭� u_{3} v_{2}) + u_{2} (u_{3} v_{1} 鈭� u_{1} v_{3}) + u_{3} (u_{1} v_{2} 鈭� u_{2} v_{1}) u 路 (u 脳 v) = 0$

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

$v 路 (u 脳 v) = (v_{1}, v_{2}, v_{3}) 路 (u_{2} v_{3} 鈭� u_{3} v_{2}, u_{3} v_{1} 鈭� u_{1} v_{3}, u_{1} v_{2} 鈭� u_{2} v_{1}) v 路 (u 脳 v) = v_{1} (u_{2} v_{3} 鈭� u_{3} v_{2}) + v_{2} (u_{3} v_{1} 鈭� u_{1} v_{3}) + v_{3} (u_{1} v_{2} 鈭� u_{2} v_{1}) v 路 (u 脳 v) = 0$

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91影视

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

Short Answer

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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91影视

If u and v are nonzero vectors in 鈩�3, why do the equations role="math" localid="1649263352081" u路(u脳v)=0 and v路(u脳v)=0 tell us that the cross product is orthogonal to both u and v?

Short Answer

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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Geometry

Mechanics Maths

Logic and Functions

Calculus

Study anywhere. Anytime. Across all devices.

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