2 questions related to mathematical vectors

[AmrAmin]

I hope I can find a solutions for those questions with your help.

1. If you know that: A=(5,-4,3) B=(2,3,-1) C=(-3,2,5) .Find the volume of the prism whose sides are $$\underline{OA}$$, $$\underline{OB}$$, $$\underline{OC}$$ .

2. Prove using vector components that:

$$\underline{a}$$ $$\times$$ $$\left($$$$\underline{b}$$ $$\times$$ $$\underline{c}$$) = ($$\underline{a}$$ . $$\underline{c}$$)$$\underline{b}$$ - ($$\underline{a}$$ . $$\underline{b}$$)$$\underline{c}$$
and using this rule prove that :

$$\underline{a}$$ $$\times$$ ($$\underline{b}$$ $$\times$$ $$\underline{c}$$) + $$\underline{b}$$ $$\times$$ ($$\underline{c}$$ $$\times$$ $$\underline{a}$$) + $$\underline{c}$$ $$\times$$ ($$\underline{a}$$ x $$\underline{b}$$) = 0

 

[chaoseverlasting]

Well? You want to give us some ideas about where you're stuck? We're not going to do it for you. We're here to help you along.

 

[tacman]

1. To find the volume of the prism, we can use the formula V = |(\underline{OA} \times \underline{OB}) . \underline{OC}|, where \underline{OA} \times \underline{OB} is the cross product of the vectors \underline{OA} and \underline{OB}.

Using the given values, we can calculate \underline{OA} \times \underline{OB} as:

\underline{OA} \times \underline{OB} = (5,-4,3) \times (2,3,-1) = (-4,17,23)

Then, substituting this into the formula, we get:

V = |(-4,17,23) . (-3,2,5)| = |(-12,-34,11)| = \sqrt{12^2 + 34^2 + 11^2} = \sqrt{1553}

Therefore, the volume of the prism is \sqrt{1553} cubic units.

2. To prove the given vector identity, we can use the fact that the cross product is distributive, meaning that:

\underline{a} \times (\underline{b} \times \underline{c}) = (\underline{a} \times \underline{b}) \times \underline{c} - (\underline{a} \times \underline{c}) \times \underline{b}

Using this, we can rewrite the left side of the given identity as:

\underline{a} \times (\underline{b} \times \underline{c}) = ((\underline{a} \times \underline{b}) \times \underline{c}) + ((\underline{a} \times \underline{c}) \times \underline{b})

Then, using the given identity, we can rewrite the right side as:

(\underline{a} . \underline{c})\underline{b} - (\underline{a} . \underline{b})\underline{c} = ((\underline{a} \times \underline{b}) \times \underline{c}) + ((\underline{a} \times \underline{c}) \times \underline{b})

Since both sides are equal, the given identity is proven.

To prove the second part, we can use the same distributive property and the given identity to rewrite the expression as: