What is the Volume of a Parallelepiped with Given Sides and Unknown Constant?

[teng125]

Find the volume of the parallelepiped depending on λ with

a=[2a,2 ,2] b=[4,1,a] c=[2,2,a] where a=[-3,1]


the [-3,1] is it let says (x,y,z) represents x and z for my substitution to find volume??
since there is no variable in y??

 

[teng125]

somebody pls help

 

[Office_Shredder]

teng, is this the direct copy of the question from the book? Because I can't figure out what the 'a' term is supposed to represent... especially since, given the context in the coordinates of A, B, and C, I would expect 'a' to be a number

 

[teng125]

a is equals to 'lamda' sign

 

[HallsofIvy]

Find the volume of the parallelepiped depending on λ with

a=[2a,2 ,2] b=[4,1,a] c=[2,2,a] where a=[-3,1]


the [-3,1] is it let says (x,y,z) represents x and z for my substitution to find volume??
since there is no variable in y??

You've actually used a in 3 different ways! I can guess that the "a" inside the brackets should really be $\lambda$ but then what does "a= [-3,1] mean??

Please don't make people guess what you mean.

My first guess at your notation is that [2a,2,2], [4,1,a], and [2,2,a] are vectors with the direction and length of the concurrent sides of the parallelopiped. If that is what is meant, please say so!

I have no guess at all as to what "a= [-3,1]" could mean. Do you know the "triple product" of vectors? This problem looks made for that.

If you don't, do you know that the cross product of two vectors, u, v, has length $|u||v| sin(\theta)$ where $\theta$ is the angle between u and v? That happens to be the area of a parallelogram with sides u and v. The volume of a parallelopiped, with three intersecting sides given by a, b, c is equal to $|a X b|\cdot c$.

 

[teng125]

[I have no guess at all as to what "a= [-3,1]" could mean.]

it means lamda vector(the sign looks like summation)[-3;1]

[My first guess at your notation is that [2a,2,2], [4,1,a], and [2,2,a] are vectors with the direction and length of the concurrent sides of the parallelopiped. If that is what is meant, please say so! ]

ya,this is the meaning.

to find the volume, i use a(vector) . [b(vector) x c(vector) ]

but,the problem is the lamda which i don't know how to get the number to substitute in order to get the volume in integer form

 

[teng125]

somebody pls help

 

[Office_Shredder]

teng, we can't help if we don't know what the question is asking

 

[StatusX]

it means lamda vector(the sign looks like summation)[-3;1]

What?? If lambda is a vector, why does it only have two components? And what could "the sign looks like summation" mean?

You are still not making sense. Please go back to your book and copy the problem exactly as written. And if there is a variable like 'lambda' in the book which you don't know how to type, do not substitute 'a' for it in your post (which you have already used in two different places!), but use L or write out lambda. And differentiate between scalars and vectors.

 

[Integral]

Teng125,
It would be worth your time, and ours, if you would learn to use our LaTex equation language.

https://www.physicsforums.com/showthread.php?t=8997"

You can simply click on any LaTex equation or symbol that apears in a post to see the code that generates it.

 

[HallsofIvy]

If three concurrent sides of the parallelpiped are given by
a=[2a,2 ,2] b=[4,1,a] c=[2,2,a] , then a ($\lambda$) can't be a vector- you can't have vector as a component of another vector.

The "triple product" of three vectors is the determinant
$$\left| \begin{array}{ccc}x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{array} \right|$$
and is equal to the product $\vec{u} \cdot \vec{v} X \vec{w}$

In this case, that would be
$$\left| \begin{array}{ccc} 2\lambda & 2 & 2 \\ 4 & 1 & \lambda \\ 2 & 2 & \lambda \end{array} \right|$$

I still don't understand what you mean by "a= [-3, 1]" since $\lambda$ must be a number, not a vector, and, anyway, a vector in this problem would have to have three components, not 2. Is it possible that you are asked to do this problem twice, once with $\lamba= -3$ and once with $\lamba= 1$?