Question

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 .

Answer 1

Answer:

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.

Answer 2

Following are solutions to the given points:

1. This dot product of a scalar and a vector, $(a\cdot b) \cdot c$, has no significance.
2. Since it is a scalar multiple of a vector, $(a \cdot b)c$  has significance.
3. Since it is the sum of two scalars, $|a|(b \cdot c)$ has meaning.
4. This dot product of two vectors, $a \cdot (b + c)$, has meaning.
5. Since it is the product of a scalar and a vector, $a \cdot b + c$ has no significance.
6. Since it is the dot product of a scalar as well as a vector, the expression $|a|\cdot (b + c)$ has no meaning.

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