Q. 77 Prove that, for vectors r, s, u,... [FREE SOLUTION]

Chapter 10: Q. 77 (page 826)

Prove that, for vectors r, s, u, and v in ${鈩�}^{3}$ ,role="math" localid="1650038347684" $(r 脳 s) 路 (u 脳 v) = (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u)$

Short Answer

Expert verified

Hence, we prove that $(r 脳 s) 路 (u 脳 v) = (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u)$ .

Step by step solution

Step 1. Given Information

Prove that, for vectors r, s, u, and v in ${鈩�}^{3}$ , $(r 脳 s) 路 (u 脳 v) = (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u)$

Step 2. Let r=(r1,r2,r3), s=(s1,s2,s3), u=(u1,u2,u3) and v=(v1,v2,v3)

Firstly finding the value of $r 脳 s$ .

role="math" localid="1650036273470" $r 脳 s = [\begin{matrix} i & j & k \\ r_{1} & r_{2} & r_{3} \\ s_{1} & s_{2} & s_{3} \end{matrix}] r 脳 s = i |\begin{matrix} r_{2} & r_{3} \\ s_{2} & s_{3} \end{matrix}| - j |\begin{matrix} r_{1} & r_{3} \\ s_{1} & s_{3} \end{matrix}| + k |\begin{matrix} r_{1} & r_{2} \\ s_{1} & s_{2} \end{matrix}| r 脳 s = i (r_{2} s_{3} - r_{3} s_{2}) - j (r_{1} s_{3} - r_{3} s_{1}) + k (r_{1} s_{2} - r_{2} s_{1}) r 脳 s = (r_{2} s_{3} - r_{3} s_{2}, r_{1} s_{3} - r_{3} s_{1}, r_{1} s_{2} - r_{2} s_{1})$

Step 3. Now finding the value of u×v

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

Step 4. Now finding the value of (r×s)·(u×v)

$(r 脳 s) 路 (u 脳 v) = (r_{2} s_{3} - r_{3} s_{2}, r_{1} s_{3} - r_{3} s_{1}, r_{1} s_{2} - r_{2} s_{1}) 路 (u_{2} v_{3} - u_{3} v_{2}, u_{1} v_{3} - u_{3} v_{1}, u_{1} v_{2} - u_{2} v_{1}) (r 脳 s) 路 (u 脳 v) = (r_{2} s_{3} - r_{3} s_{2}) 路 (u_{2} v_{3} - u_{3} v_{2}) + (r_{1} s_{3} - r_{3} s_{1}) 路 (u_{1} v_{3} - u_{3} v_{1}) + (r_{1} s_{2} - r_{2} s_{1}) 路 (u_{1} v_{2} - u_{2} v_{1}) (r 脳 s) 路 (u 脳 v) = r_{2} s_{3} (u_{2} v_{3} - u_{3} v_{2}) - r_{3} s_{2} (u_{2} v_{3} - u_{3} v_{2}) + r_{1} s_{3} (u_{1} v_{3} - u_{3} v_{1}) - r_{3} s_{1} (u_{1} v_{3} - u_{3} v_{1}) + r_{1} s_{2} (u_{1} v_{2} - u_{2} v_{1}) - r_{2} s_{1} (u_{1} v_{2} - u_{2} v_{1}) (r 脳 s) 路 (u 脳 v) = r_{2} s_{3} u_{2} v_{3} - r_{2} s_{3} u_{3} v_{2} - r_{3} s_{2} u_{2} v_{3} + r_{3} s_{2} u_{3} v_{2} + r_{1} s_{3} u_{1} v_{3} - r_{1} s_{3} u_{3} v_{1} - r_{3} s_{1} u_{1} v_{3} + r_{3} s_{1} u_{3} v_{1} + r_{1} s_{2} u_{1} v_{2} - r_{1} s_{2} u_{2} v_{1} - r_{2} s_{1} u_{1} v_{2} + r_{2} s_{1} u_{2} v_{1} (r 脳 s) 路 (u 脳 v) = (r_{2} s_{3} u_{2} v_{3} + r_{3} s_{2} u_{3} v_{2} + r_{1} s_{3} u_{1} v_{3} + r_{3} s_{1} u_{3} v_{1} + r_{1} s_{2} u_{1} v_{2} + r_{2} s_{1} u_{2} v_{1}) - (r_{2} s_{3} u_{3} v_{2} + r_{3} s_{2} u_{2} v_{3} + r_{1} s_{3} u_{3} v_{1} + r_{3} s_{1} u_{1} v_{3} + r_{1} s_{2} u_{2} v_{1} + r_{2} s_{1} u_{1} v_{2})$

Step 5. Now finding the value of r·u

$r 路 u = (r_{1}, r_{2}, r_{3}) 路 (u_{1}, u_{2}, u_{3}) r 路 u = (r_{1} u_{1} + r_{2} u_{2} + r_{3} u_{3})$

Now finding the value of $s 路 v$

role="math" localid="1650037294978" $s 路 v = (s_{1}, s_{2}, s_{3}) 路 (v_{1}, v_{2}, v_{3}) s 路 v = (s_{1} v_{1} + s_{2} v_{2} + s_{3} v_{3})$

Now finding the value of $(r 路 u) (s 路 v)$

role="math" localid="1650038075755" $(r 路 u) (s 路 v) = (r_{1} u_{1} + r_{2} u_{2} + r_{3} u_{3}) (s_{1} v_{1} + s_{2} v_{2} + s_{3} v_{3}) (r 路 u) (s 路 v) = r_{1} u_{1} (s_{1} v_{1} + s_{2} v_{2} + s_{3} v_{3}) + r_{2} u_{2} (s_{1} v_{1} + s_{2} v_{2} + s_{3} v_{3}) + r_{3} u_{3} (s_{1} v_{1} + s_{2} v_{2} + s_{3} v_{3}) (r 路 u) (s 路 v) = r_{1} u_{1} s_{1} v_{1} + r_{1} u_{1} s_{2} v_{2} + r_{1} u_{1} s_{3} v_{3} + r_{2} u_{2} s_{1} v_{1} + r_{2} u_{2} s_{2} v_{2} + r_{2} u_{2} s_{3} v_{3} + r_{3} u_{3} s_{1} v_{1} + r_{3} u_{3} s_{2} v_{2} + r_{3} u_{3} s_{3} v_{3}$

Step 5. Now finding the value of r·v

$r 路 v = (r_{1}, r_{2}, r_{3}) 路 (v_{1}, v_{2}, v_{3}) r 路 v = (r_{1} v_{1} + r_{2} v_{2} + r_{3} v_{3})$

Now finding the value of $s 路 u$

$s 路 u = (s_{1}, s_{2}, s_{3}) 路 (u_{1}, u_{2}, u_{3}) s 路 u = (s_{1} u_{1} + s_{2} u_{2} + s_{3} u_{3})$

Now finding the value of $(r 路 v) (s 路 u)$

role="math" localid="1650038126363" $(r 路 v) (s 路 u) = (r_{1} v_{1} + r_{2} v_{2} + r_{3} v_{3}) (s_{1} u_{1} + s_{2} u_{2} + s_{3} u_{3}) (r 路 v) (s 路 u) = r_{1} v_{1} (s_{1} u_{1} + s_{2} u_{2} + s_{3} u_{3}) + r_{2} v_{2} (s_{1} u_{1} + s_{2} u_{2} + s_{3} u_{3}) + r_{3} v_{3} (s_{1} u_{1} + s_{2} u_{2} + s_{3} u_{3}) (r 路 v) (s 路 u) = r_{1} v_{1} s_{1} u_{1} + r_{1} v_{1} s_{2} u_{2} + r_{1} v_{1} s_{3} u_{3} + r_{2} v_{2} s_{1} u_{1} + r_{2} v_{2} s_{2} u_{2} + r_{2} v_{2} s_{3} u_{3} + r_{3} v_{3} s_{1} u_{1} + r_{3} v_{3} s_{2} u_{2} + r_{3} v_{3} s_{3} u_{3}$

Step 7. Now finding the value of (r·u)(s·v)−(r·v)(s·u)

$(r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u) = r_{1} u_{1} s_{1} v_{1} + r_{1} u_{1} s_{2} v_{2} + r_{1} u_{1} s_{3} v_{3} + r_{2} u_{2} s_{1} v_{1} + r_{2} u_{2} s_{2} v_{2} + r_{2} u_{2} s_{3} v_{3} + r_{3} u_{3} s_{1} v_{1} + r_{3} u_{3} s_{2} v_{2} + r_{3} u_{3} s_{3} v_{3} - (r_{1} v_{1} s_{1} u_{1} + r_{1} v_{1} s_{2} u_{2} + r_{1} v_{1} s_{3} u_{3} + r_{2} v_{2} s_{1} u_{1} + r_{2} v_{2} s_{2} u_{2} + r_{2} v_{2} s_{3} u_{3} + r_{3} v_{3} s_{1} u_{1} + r_{3} v_{3} s_{2} u_{2} + r_{3} v_{3} s_{3} u_{3}) (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u) = r_{1} u_{1} s_{1} v_{1} + r_{1} u_{1} s_{2} v_{2} + r_{1} u_{1} s_{3} v_{3} + r_{2} u_{2} s_{1} v_{1} + r_{2} u_{2} s_{2} v_{2} + r_{2} u_{2} s_{3} v_{3} + r_{3} u_{3} s_{1} v_{1} + r_{3} u_{3} s_{2} v_{2} + r_{3} u_{3} s_{3} v_{3} - r_{1} v_{1} s_{1} u_{1} - r_{1} v_{1} s_{2} u_{2} - r_{1} v_{1} s_{3} u_{3} - r_{2} v_{2} s_{1} u_{1} - r_{2} v_{2} s_{2} u_{2} - r_{2} v_{2} s_{3} u_{3} - r_{3} v_{3} s_{1} u_{1} - r_{3} v_{3} s_{2} u_{2} - r_{3} v_{3} s_{3} u_{3} (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u) = (r_{2} s_{3} u_{2} v_{3} + r_{3} s_{2} u_{3} v_{2} + r_{1} s_{3} u_{1} v_{3} + r_{3} s_{1} u_{3} v_{1} + r_{1} s_{2} u_{1} v_{2} + r_{2} s_{1} u_{2} v_{1}) - (r_{2} s_{3} u_{3} v_{2} + r_{3} s_{2} u_{2} v_{3} + r_{1} s_{3} u_{3} v_{1} + r_{3} s_{1} u_{1} v_{3} + r_{1} s_{2} u_{2} v_{1} + r_{2} s_{1} u_{1} v_{2})$

Hence, prove that $(r 脳 s) 路 (u 脳 v) = (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u)$

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Prove that, for vectors r, s, u, and v in ${鈩�}^{3}$ ,role="math" localid="1650038347684" $(r 脳 s) 路 (u 脳 v) = (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u)$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Let r=(r1,r2,r3), s=(s1,s2,s3), u=(u1,u2,u3) and v=(v1,v2,v3)

Step 3. Now finding the value of u×v

Step 4. Now finding the value of (r×s)·(u×v)

Step 5. Now finding the value of r·u

Step 5. Now finding the value of r·v

Step 7. Now finding the value of (r·u)(s·v)−(r·v)(s·u)

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91影视

Prove that, for vectors r, s, u, and v in 鈩�3,role="math" localid="1650038347684" (r脳s)路(u脳v)=(r路u)(s路v)鈭�(r路v)(s路u)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Let r=(r1,r2,r3), s=(s1,s2,s3), u=(u1,u2,u3) and v=(v1,v2,v3)

Step 3. Now finding the value of u×v

Step 4. Now finding the value of (r×s)·(u×v)

Step 5. Now finding the value of r·u

Step 5. Now finding the value of r·v

Step 7. Now finding the value of (r·u)(s·v)−(r·v)(s·u)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Applied Mathematics

Logic and Functions

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Prove that, for vectors r, s, u, and v in ${鈩�}^{3}$ ,role="math" localid="1650038347684" $(r 脳 s) 路 (u 脳 v) = (r 路 u) (s 路 v) 鈭� (r 路 v) (s 路 u)$