Vector Math format question

[mcmahanrf]

I got the question:

a, b and c are three vectors such that c is perpendicular to both a and b. What is the value of a × b × c?
a. (1, 1, 1)
b. (0, 0, 0)
c. (1, 1, 0)
d. (0, 0, 1)

can someone explain what this question is looking for? I have no idea what the answers even represent. It seems like there should be more information in this question to me

[Kalbi_Rob]

I got the question:

a, b and c are three vectors such that c is perpendicular to both a and b. What is the value of a × b × c?
a. (1, 1, 1)
b. (0, 0, 0)
c. (1, 1, 0)
d. (0, 0, 1)

can someone explain what this question is looking for? I have no idea what the answers even represent. It seems like there should be more information in this question to me

a × b × c = 0

Step-by-step explanation:

Given that,

• c and a are perpendicular to each other,

then c * a = 0

• c and b are perpendicular to each other,

then c * b = 0

Now, a × b × c

= (a × b) × c

= c × (a × b)

= {(c * b)a - (c * a)b}

= {(0)a - (0)b} = 0-0 = 0

or known as vector point (0, 0, 0)

https://tardigrade.in/question/if-a-is-vector-perpendicular-to-both-b-and-c-then-vs3wj0g6

[Kalbi_Rob]

I got the question:

a, b and c are three vectors such that c is perpendicular to both a and b. What is the value of a × b × c?
a. (1, 1, 1)
b. (0, 0, 0)
c. (1, 1, 0)
d. (0, 0, 1)

can someone explain what this question is looking for? I have no idea what the answers even represent. It seems like there should be more information in this question to me

That last post probably didn't help, so I'll direct you to our favorite website, Wikipedia:

https://en.wikipedia.org/wiki/Cross_product

Specifically look at the Definition equation:

a x b = ||a|| ||b|| sin (@) * n

where @ is the angle between a and b in the plane containing them (hence, it is between 0° and 180°)
and n is a unit vector perpendicular to the plane containing a and b, in the direction given by the right-hand rule

[mcmahanrf]

a × b × c = 0

Step-by-step explanation:

Given that,

• c and a are perpendicular to each other,

then c * a = 0

• c and b are perpendicular to each other,

then c * b = 0

Now, a × b × c

= (a × b) × c

= c × (a × b)

= {(c * b)a - (c * a)b}

= {(0)a - (0)b} = 0-0 = 0

or known as vector point (0, 0, 0)

https://tardigrade.in/question/if-a-is-vector-perpendicular-to-both-b-and-c-then-vs3wj0g6


I found an easy way to figure this out, also there is a small error in your description.

The cross product function is not communitive so:

= (a × b) × c

= c × (a × b)

is not true, it should be:

= (a × b) × c

= c × -(a × b)

I'm not sure how you got the rest.



/////This is what I said
However, you don't need to do any of that because from the question we know that c is perpendicular to both a and b which means that a and b have to be parallel to each other.

Since the cross product of parallel vectors is the zero vector then:

A x B x C = 0 x C

and since the cross product of a vector with the zero vector is also zero:

0 x C = 0

which is then written in coordinate notation as (0, 0, 0)

//////This is what I realized

So just because C is perpendicular to both A and B doesn't mean that A and B have to be parallel, they can be but they don't have to be. They can also be perpendicular to each other, they can also be somewhere in between. However in any case no matter how A and B are orientated to each other the cross product will be a vector extending 90 degrees from the plane they create together. The magnitude of this vector will vary based on the angle between them but it will always be 90 degrees offset from them.

Now with that thought, we know that because the cross product must be done in order, we get:

A x B x C == (a vector that is perpendicular to A and B) x (a vector we have already been told is perpendicular to A and B)

and since these to vectors are both perpendicular to the same plane they MUST be parallel to each other. And then with the knowledge that the cross product of parallel vectors is zero we again get the correct answer of (0, 0, 0)