Question

Which of the following expressions are meaningful $?$

Which are meaningless?

• A. $\mathbf{(}\mathbf{a}\mathbf{\cdot }\mathbf{b}\mathbf{)}\mathbf{\cdot }\mathbf{c}$
• B. $\left|a\right|\mathbf{(}\mathbf{b}\mathbf{\cdot }\mathbf{c}\mathbf{)}$
• C. $\mathbf{a}\mathbf{\cdot }\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}$
• D. $\mathbf{a}\mathbf{\cdot }\mathbf{b}\mathbf{+}\mathbf{c}$
• E. $\mathbf{\left|}\mathbf{a}\mathbf{\right|}\mathbf{\cdot }\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}$

Answer 1

Answer: (B), (C)

$\mathbf{a}\mathbf{\cdot }\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}$

The explanation for the meaningful expression:

$\left(\mathrm{b}\right)\mathbf{}\mathbf{\left|}\mathbf{a}\mathbf{\right|}\mathbf{(}\mathbf{b}\mathbf{\cdot }\mathbf{c}\mathbf{)}$

The expression $\left|a\right|\mathbf{(}\mathbf{b}\mathbf{\cdot }\mathbf{c}\mathbf{)}$ has meaning because it is the product of two scalars $\left|\text{a}\right|$ and $\mathbf{b}\mathbf{\cdot }\mathbf{c}$.

Explanation: $\left|\text{a}\right|$ and $\mathbf{b}\mathbf{\cdot }\mathbf{c}$ are two scalars.

So, the product of two scalars is possible.

$\left(\mathrm{c}\right)\mathbf{a}\mathbf{\cdot }\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}$

The expression $\mathbf{a}\mathbf{\cdot }\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}$ has meaning because it is the dot product of two vectors $\mathbf{a}$ and $\mathbf{b}\mathbf{+}\mathbf{c}$.

Explanation: The sum of two vectors $\mathbf{a}$ and $\mathbf{b}\mathbf{+}\mathbf{c}$ is a vector.

Hence, the dot product of two vectors is possible.

The explanation for the meaningless expression.

$\left(\mathrm{a}\right)$ $\mathbf{(}\mathbf{a}\mathbf{\cdot }\mathbf{b}\mathbf{)}\mathbf{\cdot }\mathbf{c}$

The expression $\mathbf{(}\mathbf{a}\mathbf{\cdot }\mathbf{b}\mathbf{)}\mathbf{\cdot }\mathbf{c}$ has no meaning because, it is the dot product of a scalar $\mathbf{a}\mathbf{\cdot }\mathbf{b}$ and a vector $\mathbf{c}$.

Explanation: The dot product $\mathbf{a}\mathbf{\cdot }\mathbf{b}$ is scalar and $\mathbf{c}$ is a vector, and the product of a scalar and a vector is not possible.

$\left(\mathrm{d}\right)\mathbf{a}\mathbf{\cdot }\mathbf{b}\mathbf{+}\mathbf{c}$

The expression $\mathbf{a}\mathbf{\cdot }\mathbf{b}\mathbf{+}\mathbf{c}$ has no meaning because it is the sum of a scalar $\mathbf{a}\mathbf{\cdot }\mathbf{b}$ and a vector $\mathbf{c}$.

Explanation: The addition of a scalar $\mathbf{a}\mathbf{\cdot }\mathbf{b}$ and a vector $\mathbf{c}$ is not possible as they have a different dimensions.

$\left(e\right)\left|a\right|\mathbf{\cdot }\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}$

The expression $\left|a\right|\mathbf{\cdot }\mathbf{(}\mathbf{b}\mathbf{+}\mathbf{c}\mathbf{)}$ has no meaning because it is the dot product of a scalar $\left|a\right|$ and a vector $\mathbf{b}\mathbf{+}\mathbf{c}$.

Explanation: Dot product has meaning only for pairs of vectors.

Hence, the dot product of a scalar and a vector is not possible.