Compute some vector algebra expressions

Let

$A = (2,4,-7), \qquad B = (2,6,3), \qquad C = (3,4,-5).$

Compute each of the following (and insert appropriate parentheses to make sensible expressions):

First, we have $A \cdot BC$ means $(A \cdot B)C$ , then we compute
$\begin{align*} (A \cdot B)C &= ((2,4,-7) \cdot (2,6,3))(3,4,-5) \\ &= ((2)(2) + (4)(6)+ (-7)(3)) (3,4,-5)\\ &= (4 + 24 - 21)(3,4,-5)\\ &= 7(3,4,-5)\\ &= (21,28,-35). \end{align*}$
First, we have $A \cdot B + C$ means $A \cdot (B+C)$ , then we compute
$\begin{align*} A \cdot (B+C) &= (2,4,-7) \cdot ((2,6,3) +(3,4,-5)) \\ &= (2,4,-7) \cdot (5,10,-2)\\ &= (2)(5) + (4)(10) + (-7)(-2) \\ &= 10 + 40 + 14\\ &= 64. \end{align*}$
First, we have $A + B \cdot C$ means $(A+B)\cdot C$ , then we compute
$\begin{align*} (A+B) \cdot C &= ((2,4,-7) + (2,6,3)) \cdot (3,4,-5) \\ &= (4,10,-4) \cdot (3,4,-5)\\ &= (4)(3)+ (10)(4) + (-4)(-5) \\ &= 12 + 40 + 20 \\ &= 72. \end{align*}$
First, we have $AB \cdot C$ means $A(B \cdot C)$ , then we compute
$\begin{align*} A(B \cdot C) &= (2,4,-7)((2,6,3) \cdot (3,4,-5)) \\ &= (2,4,-7)((2)(3) + (6)(4) + (3)(-5)) \\ &= (2,4,-7)(6 + 24 - 15)\\ &= (2,4,-7)(15)\\ &= (30,60,-105). \end{align*}$
First, we have $A/B\cdot C$ means $A / (B \cdot C)$ , then we compute
$\begin{align*} A / (B \cdot C) &= (2,4,-7) / ((2,6,3) \cdot (3,4,-5)) \\ &= (2,4,-7) / ((2)(3) + (6)(4) + (3)(-5)) \\ &= (2,4,-7) / (6 + 24 - 15) \\ &= (2,4,-7) / (15) \\ &= \left( \frac{2}{15}, \frac{4}{15}, -\frac{7}{15} \right). \end{align*}$

Stumbling Robot