 # Which of the following expressions are meaningful which are meaningless explain: 1. (a . b) . c
2. (a . b) c
3. |a|(b . c)
4. a . (b + c)
5. a . b + c
6. |a| . (b+c)

The questions aim to find the expressions of some vector multiplication and addition to check whether the expression is meaningful or meaningless.

The background concept needed for this question to solve include scalar addition and multiplication, vector addition and multiplication, and addition and multiplication of the vector magnitude.

By using the properties of Scalar and Vector, we have to find wheater the given expressions are meaningful or meaningless.

a) $(a.b).c$

The given expression shows that it is a dot product of two scalars $a$ and $b$ to the vector $c$ which is not a meaningful expression.

b) $(a.b)c$

The given expression shows that it is a dot product of two scalars $a$ and $b$ which will result in a scalar and we can multiply it to the vector $c$ which is significant and means that the given expression is meaningful.

c) $|a|(b . c)$

The $|a|$ given expression shows that it is the magnitude of the vector and the magnitude is always scalar. The dot product of two scalars $a$ and $b$ will result in a scalar and we can multiply it to the magnitude of $|a|$  which is a scalar. So scalar can be multiplied with the scalar and this results in that the given expression is meaningful.

d) $a.(b + c)$

The $(b+c)$ in the given expression results in a vector which shows that it is an addition of $a$ and $b$. Now we can take the scalar product of a vector with the other vector $c$. So the given equation is significant which means that it is not meaningless.

e) $a.b+c$

The dot product of $a.b$ in the given expression will result in a scalar and thus we can not add it to the vector $c$. Hence the addition of vector and scalar is not possible. So the given expression is not significant which means it is not meaningful.

f) $|a|.(b+c)$

The $|a|$ given expression shows that it is the magnitude of the vector and the magnitude is always scalar. The $(b+c)$ in the given expression will result in a vector. So dot product of a scalar with a vector is not possible which shows that the given expression is not significant and means that it is not meaningful.

By using the  concept of scalar addition and multiplication, vector addition and multiplication, and addition and multiplication of the vector magnitude, it is shown that:

The given expression $(a . b). c$ is not a meaningful expression.

The given expression $(a . b)c$ is a meaningful expression.

The given expression $|a|(b . c)$ is a meaningful expression.

The given expression $a.(b + c)$ is not meaningless expression.

The given expression $a.b+c$  is not meaningful expression.

The given expression $|a|.(b+c)$  is not meaningful expression.

## Example

Show that the given expression $(x.y).z^2$ is a meaningful or meaningless expression.

The given expression $(x.y).z^2$ shows that it is a dot product of two scalars $x$ and $y$  and $z^2$ shows a scalar as squaring a vector will result in a scalar. Thus the given expression is significant which means that it is a meaningful expression.