+0

# Let be a vector of length 2, and let and be vectors such that Find . If any of these cannot be uniquely determined from the information

0
58
2

Let $$\mathbf{a}$$ be a vector of length 2, and let $$\mathbf{b}$$ and $$\mathbf{c}$$ be vectors such that $$\mathbf{a} \bullet \mathbf{b} = 3, \mathbf{a} \bullet \mathbf{c} = 4, \mathbf{b} \bullet \mathbf{c} = 5.$$ Find$$\mathbf{a}\bullet( \mathbf{b} + \mathbf{c}), \mathbf{b}\bullet(\mathbf{a} +\mathbf{b} + \mathbf{c}), (\mathbf{a} + 2\mathbf{b})\bullet(-3\mathbf{a} + 3\mathbf{c})$$. If any of these cannot be uniquely determined from the information given, enter a question mark.

Could someone help or point me in the right direction? Thank you!

Feb 24, 2020

#1
+24366
+1

Let $$\mathbf{a}$$ be a vector of length 2, and let $$\mathbf{b}$$ and $$\mathbf{c}$$ be vectors such that
$$\mathbf{a} \bullet \mathbf{b} = 3,\ \mathbf{a} \bullet \mathbf{c} = 4,\ \mathbf{b} \bullet \mathbf{c} = 5$$.
Find $$\mathbf{a}\bullet( \mathbf{b} + \mathbf{c}),\ \mathbf{b}\bullet(\mathbf{a} +\mathbf{b} + \mathbf{c}),\ (\mathbf{a} + 2\mathbf{b})\bullet(-3\mathbf{a} + 3\mathbf{c})$$.
If any of these cannot be uniquely determined from the information given, enter a question mark.

$$\mathbf{a}\bullet( \mathbf{b} + \mathbf{c})\\ \begin{array}{|rcll|} \hline && \mathbf{a}\bullet( \mathbf{b} + \mathbf{c}) \\ &=& \mathbf{a}\bullet\mathbf{b}+\mathbf{a}\bullet \mathbf{c} \quad | \quad \mathbf{a} \bullet \mathbf{b} = 3,\ \mathbf{a} \bullet \mathbf{c} = 4 \\ &=& 3+4 \\ &=& \mathbf{7} \\ \hline \end{array}$$

$$\mathbf{b}\bullet(\mathbf{a} +\mathbf{b} + \mathbf{c}) \\ \begin{array}{|rcll|} \hline && \mathbf{b}\bullet(\mathbf{a} +\mathbf{b} + \mathbf{c}) \\ &=& \mathbf{b}\bullet \mathbf{a} + \mathbf{b}\bullet\mathbf{b} + \mathbf{b}\bullet\mathbf{c} \\ &=& \mathbf{a}\bullet \mathbf{b} + \mathbf{b}\bullet\mathbf{b} + \mathbf{b}\bullet\mathbf{c} \\ &=& 3 + \mathbf{b}\bullet\mathbf{b} + 5 \\ &=& 8 + \mathbf{b}\bullet\mathbf{b} \\ &=& ? \\ \hline \end{array}$$

$$(\mathbf{a} + 2\mathbf{b})\bullet(-3\mathbf{a} + 3\mathbf{c}) \\ \begin{array}{|rcll|} \hline && (\mathbf{a} + 2\mathbf{b})\bullet(-3\mathbf{a} + 3\mathbf{c}) \\ &=& (-3)\mathbf{a}\bullet\mathbf{a}+3\mathbf{a}\bullet\mathbf{c}-6\mathbf{b}\bullet\mathbf{a}+6\mathbf{b}\bullet\mathbf{c} \\ &=& (-3)\mathbf{a}\bullet\mathbf{a}+3\mathbf{a}\bullet\mathbf{c}-6\mathbf{a}\bullet\mathbf{b}+6\mathbf{b}\bullet\mathbf{c} \quad | \quad \mathbf{a}\bullet\mathbf{a} = 2^2 \\ &=& (-3)*2^2+3*4-6*3+6*5 \\ &=& -12+12-18+30 \\ &=& -18+30 \\ &=& \mathbf{12} \\ \hline \end{array}$$

Feb 25, 2020
#2
0

Thank you so much for your answer, it really helped me!! Really appreciate it :)

Guest Mar 1, 2020