How to find the volume of a parallelepiped using determinants?

[ineedhelpnow]

find the parallelepiped determined by the vector $a= \left\langle 1,2,3 \right\rangle$, $b= \left\langle -1,1,2 \right\rangle$, $c= \left\langle 2,1,4 \right\rangle$

find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. $P(-2,1,0)$, $Q(2,3,2)$, $R(1,4,-1)$, $S(3,6,1)$

1. can some please explain what a parallelepiped is
2. how to find the volume of one (i don't need to see it worked but steps on what to do what be really helpful)
3. how is the second question similar to the first

thanks. there are no examples like this in my book or notes

 

[ineedhelpnow]

ok so for the first one i managed to find this equation $V= \left| a \cdot (b \times c) \right|$
so i get that one, but what about the second problem. is it similar to this?

 

[Euge]

ok so for the first one i managed to find this equation $V= \left| a \cdot (b \times c) \right|$
so i get that one, but what about the second problem. is it similar to this?

Yes, it is similar. When you're given two points in three-space, say $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, you can find the vector from $A$ to $B$ (denoted by $\vec{AB}$) by subtracting each component of $B$ from the corresponding component from $A$. 2
$\displaystyle \vec{AB}\, =\, <x_2 - x_1, y_2 - y_1, z_2 - z_1> $.

Using this formula, we find

$\displaystyle \vec{PQ}\, =\, <2 - (-2), 3 - 1, 2 - 0>\, =\, <4, 2, 2>$,

$\displaystyle \vec{PR}\, =\, <1 - (-2), 4 - 1, 2 - 0>\, =\, <3, 3, 2>$,

$\displaystyle \vec{PS}\, =\, <3 - (-2), 6 - 1, 1 - 0>\, =\, <5, 5, 1>$.

Now you can compute the volume of the parallelopiped: $|\vec{PQ} \cdot (\vec{PR} \times \vec{PS})|$.

 

[Prove It]

ok so for the first one i managed to find this equation $V= \left| a \cdot (b \times c) \right|$
so i get that one, but what about the second problem. is it similar to this?

Actually since $\displaystyle \begin{align*} \mathbf{a} \cdot \mathbf{b} \times \mathbf{c} \end{align*}$ is a scalar value, you don't need to take its magnitude (you can but it's a pointless extra step)...

 

[ineedhelpnow]

$\displaystyle \vec{PQ}\, =\, <2 - (-2), 3 - 1, 2 - 0>\, =\, <4, 4, 2>$,

thanks! i never knew that 3-1 is equal to four :p

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Actually since $\displaystyle \begin{align*} \mathbf{a} \cdot \mathbf{b} \times \mathbf{c} \end{align*}$ is a scalar value, you don't need to take its magnitude (you can but it's a pointless extra step)...

i see. i think I am going to make a note of that in my homework so he doesn't get confused. :)

 

[Euge]

thanks! i never knew that 3-1 is equal to four :p

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i see. i think I am going to make a note of that in my homework so he doesn't get confused. :)

Sorry for the typo -- I've fixed it. You do need the absolute value bars in the formula for the volume, since the scalar triple product may be negative. It is not a pointless step.

 

[ineedhelpnow]

so it is absolute value? that's what i thought at first but then for some reason it made more sense when he said it was magnitude because this chapter's all about magnitudes and stuff.

but i found in my book that is says "the volume of the parallelepiped determined by the vectors is the MAGNITUDE of their scalar triple product." so i think PROVE IT is right

 

[Euge]

The magnitude of a real number and the absolute value of a real number are the same.

 

[Prove It]

thanks! i never knew that 3-1 is equal to four :p

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i see. i think I am going to make a note of that in my homework so he doesn't get confused. :)

Performing products on vectors will become a disaster if you don't at least have a decent idea of what SORT of answer to expect.

Type 1: Magnifying a vector by a scalar value, e.g, $\displaystyle \begin{align*} a\mathbf{b} \end{align*}$ is written without ANY "product symbol", expect a VECTOR for the answer.

Type 2: The DOT PRODUCT of two vectors, e.g. $\displaystyle \begin{align*} \mathbf{a} \cdot \mathbf{b} \end{align*}$, defined as the sum of the products of the corresponding components of the two vectors, expect a SCALAR for the answer. (Sometimes this is called the scalar product for that reason).

Type 3: The CROSS PRODUCT of two vectors, e.g. $\displaystyle \begin{align*} \mathbf{a} \times \mathbf{b} \end{align*}$ is defined as finding a vector that is normal to both vectors a and b. Thus you expect a VECTOR as your answer. (Sometimes this is called the vector product for that reason).

Thus when you have a more difficult looking product, you first need to think about "what is the order of the products which must be used?" (some will be obvious, like the scalar triple product $\displaystyle \begin{align*} \mathbf{a} \cdot \mathbf{b} \times \mathbf{c} \end{align*}$, some will be not obvious, like the vector triple product $\displaystyle \begin{align*} \mathbf{a} \times \left( \mathbf{b} \times \mathbf{c} \right) \end{align*}$ ), then you think about what sort of answer you should be getting (a vector or a scalar), then perform any simplification (e.g. there is an identity which makes calculating a vector triple product much easier - you will get to this later), then finally actually perform the calculations.As for needing the absolute value sign, the negative value should be telling the reader something about the positioning of the paralleliped, the volume should still be immediately obvious.Magnitude and Absolute Value essentially mean the same thing (the size of a quantity), it's just that these quantities may be of different types.

 

[ineedhelpnow]

(Tmi) i did not know that the mag and abs. value of a real number were the same. thank you for clarifying that.

and thank you for your explanation prove it. i don't mean to seem dumb when i ask questions (after all i am one of the very few that got an A in my calc 2 class over the summer ;) ) but I am still trying to figure this whole subject out :) (i mean this in a kind way)

 

[Evgeny.Makarov]

You can also use determinants to find volume. If $\vec{a}=(a_1,a_2,a_3)$, $\vec{b}=(b_1,b_2,b_3)$ and $\vec{c}=(c_1,c_2,_3)$, then the volume of the parallelepiped defined by $\vec{a}$, $\vec{b}$ and $\vec{c}$ is
\[
\left| \det \begin{pmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{pmatrix} \right|.
\]

If a parallelepiped is given by four vertices $P(p_1,p_2,p_3)$, $Q(q_1,q_2,q_3)$, $R(r_1,r_2,r_3)$ and $S(s_1,s_2,s_3)$ where $Q$, $R$ and $S$ are adjacent to $P$, its volume is
\[
\left| \det \begin{pmatrix}
p_1 & p_2 & p_3 & 1\\
q_1 & q_2 & q_3 & 1\\
r_1 & r_2 & r_3 & 1\\
s_1 & s_2 & s_3 & 1
\end{pmatrix} \right|.
\]