# Which of the following expressions are meaningful? Which are meaningless? Explain. a) (a\cdot b)\cdot c (a\cdot b)\cdot c has ? because it is the dot

Which of the following expressions are meaningful? Which are meaningless? Explain.
a) $\left(a\cdot b\right)\cdot c$
$\left(a\cdot b\right)\cdot c$ has ? because it is the dot product of ?.
b) $\left(a\cdot b\right)c$
$\left(a\cdot b\right)c$ has ? because it is a scalar multiple of ?.
c) $|a|\left(b\cdot c\right)$
$|a|\left(b\cdot c\right)$ has ? because it is the product of ?.
d) $a\cdot \left(b+c\right)$
$a\cdot \left(b+c\right)$ has ? because it is the dot product of ?.
e) $a\cdot b+c$
$a\cdot b+c$ has ? because it is the sum of ?.
f) $|a|\cdot \left(b+c\right)$
$|a|\cdot \left(b+c\right)$ has ? because it is the dot product of ?.
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Step 1
a) The expression $\left(a\cdot b\right)\cdot c$ has meaningless because, it is the dot product of a scalar $a\cdot b$ and a vector c.
Note that here, the dot product $a\cdot b$ is a scalar, and c is a vector, and a scalar and a vector cannot be dot product with each other.
b) The expression $\left(a\cdot b\right)c$ has meaningful because, it is a scalar multiple of a scalar $a\cdot b$ and the vector c.
Note that here, the dot product $a\cdot b$ is a scalar, and c is a vector, and a scalar multiplication is possible with a vector.
c) The expression $|a|\left(b\cdot c\right)$ has meaningful because, it is the product of two scalars $|a|$ and $b\cdot c$.
Here, two scalars can be multiplied easily.
d) The expression $a\cdot \left(b+c\right)$ has meaningful because, it is the dot product of two vectors a and $b+c$.
Note that here, the sum of two vectors, $b+c$ is again a vector, and the two vectors dot product with each other.
e) The expression $a\cdot b+c$ has meaningful because, it is the sum of a scalar $a\cdot b$ and the vector c.
Note that, a scalar and a vector cannot be added.
f) The expression $|a|\cdot \left(b+c\right)$ has meaningless because, it is the dot product of a scalar $|a|$ and a vector $b+c$.
Note that, a scalar and a vector cannot be dot product with each other.