Which of the following are valid expressions?

Answers:

In short, the boxes you check will be box 1, box 3, box 6.

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.

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.

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

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

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.

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.

Choice 6

B x C is some vector

A dot (B x C) results in a scalar

The expression is valid

the one with the highest speed

the bal will not change its velocity.