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Q1E

Expert-verifiedFound in: Page 556

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

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

(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

**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.**

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.

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.

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.

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.

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.

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.

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