Question

To determine whether the given expression is meaningful or meaningless.

(a) \(({\rm{a}} \cdot {\rm{b}}) \cdot {\rm{c}}\)

(b) \((a \cdot b)c\)

(c) \(|{\rm{a}}|({\rm{b}} \cdot {\rm{c}})\)

(d) \(a \cdot (b + c)\)

(e) \(a \cdot b + c\)

(f) \(|a| \cdot (b + c)\)

Short answer

(a) The expression is meaningless

(b) The expression is meaningful

(c) The expression is meaningful

(d) The expression is meaningful

(e) The expression is meaningless

(f) The expression is meaningless

Step-by-step solution

Concept of Dot Product

A minimum of two vectors are required to perform a dot product. The resultant dot product of two vectors is a scalar. So, the dot product is also known as a scalar product.

Check whether the expression is meaningful or meaningless

The resultant dot product of \(({\rm{a}} \cdot {\rm{b}})\) is a scalar value. Since \(({\rm{a}} \cdot {\rm{b}})\) is a scalar value, the dot product between a scalar value \(({\rm{a}} \cdot {\rm{b}})\) and vector \((c)\)cannot be performed.

Therefore \(({\rm{a}} \cdot {\rm{b}}) \cdot {\rm{c}}\) is a meaningless expression.

Check whether the expression is meaningful or meaningless

The resultant dot product of \(({\rm{a}} \cdot {\rm{b}})\) is a scalar value and the multiplication of this scalar value \(({\rm{a}} \cdot {\rm{b}})\)with a vector \((c)\)is a vector.

Therefore \((a \cdot b)c\) is a meaningful expression.

Check whether the expression is meaningful or meaningless

Consider a general expression to find the magnitude of a two-dimensional vector, that is, \(a = \left\langle {{a_1},{a_2}} \right\rangle \).

\(|{\rm{a}}| = \sqrt {a_1^2 + a_2^2} \)

The resultant magnitude of a vector is scalar.

Since the magnitude \((|a|)\) and dot product of \((b \cdot c)\) are scalar values, the multiplication \(|a|(b \cdot c)\) is a meaningful one.

Therefore \(|a|(b \cdot c)\) is a meaningful expression.

Check whether the expression is meaningful or meaningless

A minimum of two vectors are required to perform a summation operation and the resultant value is in a vector.

Since the summation of \((b + c)\) is a vector, the dot product between a vector \((a)\) and \((b + c)\) is possible and the resultant value of \(a \cdot (b + c)\) is scalar.

Therefore \(a \cdot (b + c)\) is a meaningful expression.

Check whether the expression is meaningful or meaningless

A minimum of two vectors are required to perform a summation operation and the resultant value is in a vector.

Since \(({\rm{a}} \cdot {\rm{b}})\) is a scalar product, the additional operation between a scalar value and a vector cannot be performed.

Therefore \(a \cdot b + c\) is a meaningless expression.

Check whether the expression is meaningful or meaningless

Consider a general expression to find the magnitude of a two-dimensional vector that is\(a = \left\langle {{a_1},{a_2}} \right\rangle \)\(|{\rm{a}}| = \sqrt {a_1^2 + a_2^2} \)

The resultant magnitude of a vector is scalar.

Since the magnitude of a vector \(\left( a \right)\) is a scalar value, the dot product between a scalar value \(\left( {|a|} \right)\)and a vector \((b + c)\)cannot be performed.

Therefore \(|a| \cdot (b + c)\) is a meaningless expression.