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)

Hence, we prove that (r×s)·(u×v)=(r·u)(s·v)−(r·v)(s·u).

Q. 77 - Calculus | StudyQuestionHub

Q. 77

Question

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-by-Step Solution

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Answer

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

1Step 1. Given Information

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

2Step 2. Let r = ( r 1 , r 2 , r 3 ) ,   s = ( s 1 , s 2 , s 3 ) ,   u = ( u 1 , u 2 , u 3 )   and   v = ( v 1 , v 2 , v 3 )

Firstly finding the value of $r \times s$ .

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

3Step 3. Now finding the value of u × v

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

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

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

5Step 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 $s \cdot v$

$s \cdot v = (s_{1}, s_{2}, s_{3}) \cdot (v_{1}, v_{2}, v_{3}) s \cdot v = (s_{1} v_{1} + s_{2} v_{2} + s_{3} v_{3})$

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

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

6Step 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 $(r \cdot v) (s \cdot u)$

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

7Step 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 $(r \times s) \cdot (u \times v) = (r \cdot u) (s \cdot v) - (r \cdot v) (s \cdot u)$

Q. 76

Q. 78

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Q. 76

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