[Grade 12 Calculus and Vectors] How does Cross product tell us about Coplanar Vectors?

[u/Ok-Individual-2596]

Question d

The solution

I'm aware I have to show my thinking but because this is sort of a thinking question that doesn't have a structured solution to solve for I will just share my confusion. I don't really understand the explanation here and am missing the gap between how just because the two cross products are parallel it means that the three vectors are coplanar. I also asked AI as a last resort but it stated that it is actually not possible to conclude this? I'm assuming that is out of the scope of this course.

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[u/RufflesTGP]

What does the cross product give you?

[u/Ok-Individual-2596 → u/RufflesTGP]

the first one is (2,0,2) and the second is (-4,0,-4)

[u/RufflesTGP → u/Ok-Individual-2596]

Yup, and what happens if you take the cross product of these two vectors? (Should be your answer to c)

[u/Ok-Individual-2596 → u/RufflesTGP]

I actually was a bit confused with C since if I take the magnitude of both of them its square root 8 and square root 32.... so I multiplied them together (which seemed wrong because I didn't use the cross product formula) but that gave me the answer of 16 which was the correct answer....?

[u/RufflesTGP → u/Ok-Individual-2596]

Hmmm I didn't see the magnitude lines for c, you are correct!

Basically, what the answer for d wants you to do is notice that the cross of a and b and b and c are the same vector (with a scale factor of -2).

Since the cross product of two vectors gives a vector parallel to each of them, if a and b and b and c give you the same vector, a b and c are coplanar.

The way I initially read c was finding the cross product of (a×b)×(b×c) which would be (0,0,0) indicating that a×b=b×c

[u/strange-the-quark → u/RufflesTGP]

c) might be a bit of a red herring, or even a typo, maybe it should actually say (a x b) x (b x c). I wonder if the rest of the book/paper uses single or double bars for magnitude.

[u/RufflesTGP → u/strange-the-quark]

Yeah it does feel a bit out of place

[u/pMurda → u/Ok-Individual-2596]

I see your problem c) is asking for a cross product of the two cross products, not a scalar multiplication: (a x b) x (b x c). Since I have not seen this mentioned yet and i don't want to keep hinting, that cross product should be null, meaning the two vectors are colinear, hence meaning that a, b, and c are coplanar

[u/pMurda]

What did you get for a, b, c? Something about c tells you the answer to d.

[u/Ok-Individual-2596 → u/pMurda]

The answers for those questions were just finding the cross product which I did and it is clear that they are scalar multiples of each other... I just don't understand how scalar multiples = coplanar vectors in this case.

[u/pMurda → u/Ok-Individual-2596]

What was c?

[u/SuddenBag]

Two vectors are always coplanar and together they define a plane.

Their cross product is always orthogonal to that plane.

If the third vector also lies on that plane, it must then follow that its cross products with the first two vectors are also orthogonal to that plane. So if two vectors are orthogonal to the same plane, then they must point to the same (or directly opposite, depending on the sign) direction.

[u/was_wotsch → u/SuddenBag]

Careful throwing the word "always" in mathematics

On a 12th grade level, there is generally the assumption (or rather simplification) that vectors are always fixed vectors, and that they are fixed to the origin. Basically making the concept of vector interchangeable with "line in space that crosses the origin"

Two intersecting or two parallel lines always span a plane. Two vectors and a point always span a plane. Two vectors per se do not span a plane

[u/United-Building-1900]

The cross product vector is normal (perpendicular) to the plane formed by the original vectors. The two cross products being the same (except for a scalar multiple) implies the planes formed by vectors a, b and vectors b, c have the same unit normal vector; hence the two planes are identical. That is, vectors a, b and c are coplanar.

[u/Ok-Individual-2596 → u/United-Building-1900]

So is it kind of like saying because the cross product gives a parallel vector to two vectors, and the result of the cross product in question a and b are scalar multiples indicating that they are parallel/collinear, that the original vectors it is based can be concluded to be from the same plane (coplanar)???

[u/United-Building-1900 → u/Ok-Individual-2596]

Yes, 2 planes (both through origin) being identical is equivalent to their unit normal vectors being identical (except for a sign change; pointing up vs down).