**(a . b) . c****(a . b) c****|a|(b . c)****a . (b + c)****a . b + c****|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**.

## Expert Answer

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 produc**t 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 a**ddition 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**.

## Numerical Answer

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**.