Which of the following expressions are meaningful? Which are meaningless? Explain. (a) (a · b) · c (a · b) · c has because it is the dot product of. (b) (a · b) c (a · b) c has because it is a scalar multiple of. (c) |a| (b · c) |a| (b · c) has because it is the product of. (d) a · (b + c) a · (b + c) has because it is the dot product of. (e) a · b + c a · b + c has because it is the sum of. (f) |a| · (b + c) |a| · (b + c) has because it is the dot product of.

a. It is meaningless.

b. It has meaning.

c. It has meaning.

d. It has meaning.

e. It is meaningless.

f. It is meaningless.

Step-by-step explanation:

a. (a. b). c.

This is meaningless, a. b will give a scalar and we can't take the dot product of the scalar result with vector c.

b. (a. b) c

This has meaning, a. b will give a scalar, the the scalar will be multiply with the vector c.

c. |a| (b. c)

This has meaning, |a| means magnitude of vector a and it will a scalar and this is multiple by the scalar result of b. c

d. a. (b+c)

This has meaning, b+c will give another vector then, the scalar product of the result and vector c will be calculated and it will gives us a scalar answer.

e. a. b + c

This is meaningless, the dot product of a. b will give a scalar result and the scalar result canont be added to c. We can't add a scalar to a vector.

f. |a|. (a+b)

This is meaningless, |a| means the magnitude of vector a and that is a scalar. The scalar product of a scalar and a vector cannot be calculated.