Answers:
- valid
- not valid
- valid
- not valid
- not valid
- valid
In short, the boxes you check will be box 1, box 3, box 6.
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Explanation:
It's not stated, but I'm assuming that A,B,C are three dimensional vectors
I'll go through each answer choice to say whether it's valid or not.
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Choice 1
A x B is some vector
(A x B) x C is some other vector. Cross products produce vectors as outputs, when the two inputs are both 3D vectors. The output vector is the same dimension as the inputs.
Overall, the expression (A x B) x C is valid.
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Choice 2
A dot B is a scalar
(A dot B)C is some vector
A dot (B x C) is a scalar
We cannot add a scalar to a vector. The given expression (A dot B)C + A dot (B x C) is not valid
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Choice 3
B x C is a vector
B dot (B x C) is scalar
A[ B dot (B x C) ] is a vector in the form kA with k as a scalar.
The overall expression is valid
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Choice 4
B x C is a vector
B(B x C) is not valid. We need to use a dot or cross symbol between the B and (B x C).
If we had kB with k as a scalar, then we don't need a symbol between them.
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Choice 5
A dot B is a scalar
Scalars are not allowed when it comes to cross products, so (A dot B) crossed with anything is not valid.
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Choice 6
B x C is some vector
A dot (B x C) results in a scalar
The expression is valid
