Chapter 10: Q. 77 (page 826)

Prove that, for vectors r, s, u, and v in $ℝ^{3}$ ,role="math" localid="1650038347684" $(r \times s) \cdot (u \times v) = (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u)$

Short Answer

Expert verified

Hence, we prove that $(r \times s) \cdot (u \times v) = (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u)$ .

Step by step solution

Step 1. Given Information

Prove that, for vectors r, s, u, and v in $ℝ^{3}$ , $(r \times s) \cdot (u \times v) = (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u)$

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 $r \times s$ .

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

Step 3. Now finding the value of u×v

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

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

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

Step 5. Now finding the value of r·u

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

Now finding the value of $s \cdot v$

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

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

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

Step 5. Now finding the value of r·v

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

Now finding the value of $s \cdot u$

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

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

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

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

$(r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u) = r_{1} u_{1} s_{1} v_{1} + r_{1} u_{1} s_{2} v_{2} + r_{1} u_{1} s_{3} v_{3} + r_{2} u_{2} s_{1} v_{1} + r_{2} u_{2} s_{2} v_{2} + r_{2} u_{2} s_{3} v_{3} + r_{3} u_{3} s_{1} v_{1} + r_{3} u_{3} s_{2} v_{2} + r_{3} u_{3} s_{3} v_{3} - (r_{1} v_{1} s_{1} u_{1} + r_{1} v_{1} s_{2} u_{2} + r_{1} v_{1} s_{3} u_{3} + r_{2} v_{2} s_{1} u_{1} + r_{2} v_{2} s_{2} u_{2} + r_{2} v_{2} s_{3} u_{3} + r_{3} v_{3} s_{1} u_{1} + r_{3} v_{3} s_{2} u_{2} + r_{3} v_{3} s_{3} u_{3}) (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u) = r_{1} u_{1} s_{1} v_{1} + r_{1} u_{1} s_{2} v_{2} + r_{1} u_{1} s_{3} v_{3} + r_{2} u_{2} s_{1} v_{1} + r_{2} u_{2} s_{2} v_{2} + r_{2} u_{2} s_{3} v_{3} + r_{3} u_{3} s_{1} v_{1} + r_{3} u_{3} s_{2} v_{2} + r_{3} u_{3} s_{3} v_{3} - r_{1} v_{1} s_{1} u_{1} - r_{1} v_{1} s_{2} u_{2} - r_{1} v_{1} s_{3} u_{3} - r_{2} v_{2} s_{1} u_{1} - r_{2} v_{2} s_{2} u_{2} - r_{2} v_{2} s_{3} u_{3} - r_{3} v_{3} s_{1} u_{1} - r_{3} v_{3} s_{2} u_{2} - r_{3} v_{3} s_{3} u_{3} (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u) = (r_{2} s_{3} u_{2} v_{3} + r_{3} s_{2} u_{3} v_{2} + r_{1} s_{3} u_{1} v_{3} + r_{3} s_{1} u_{3} v_{1} + r_{1} s_{2} u_{1} v_{2} + r_{2} s_{1} u_{2} v_{1}) - (r_{2} s_{3} u_{3} v_{2} + r_{3} s_{2} u_{2} v_{3} + r_{1} s_{3} u_{3} v_{1} + r_{3} s_{1} u_{1} v_{3} + r_{1} s_{2} u_{2} v_{1} + r_{2} s_{1} u_{1} v_{2})$

Hence, prove that $(r \times s) \cdot (u \times v) = (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Prove that, for vectors r, s, u, and v in $ℝ^{3}$ ,role="math" localid="1650038347684" $(r \times s) \cdot (u \times v) = (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u)$

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

Discrete Mathematics

Logic and Functions

Probability and Statistics

Geometry

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

Company

Product

Help

Prove that, for vectors r, s, u, and v in ℝ3,role="math" localid="1650038347684" (r×s)·(u×v)=(r·u)(s·v)−(r·v)(s·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

Discrete Mathematics

Logic and Functions

Probability and Statistics

Geometry

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

Prove that, for vectors r, s, u, and v in $ℝ^{3}$ ,role="math" localid="1650038347684" $(r \times s) \cdot (u \times v) = (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u)$