Question

Which of the following expressions are meaningful? Which are meaningless? Explain.

(a) (a · b) · c has ---Select--- meaning no meaning because it is the dot product of ---Select--- a scalar and a vector two vectors two scalars .
(b) (a · b)c has ---Select--- meaning no meaning because it is a scalar multiple of ---Select--- a scalar a vector .
(c) |a|(b · c) has ---Select--- meaning no meaning because it is the product of ---Select--- a scalar and a vector two vectors two scalars .
(d) a · (b + c) has ---Select--- meaning no meaning because it is the dot product of ---Select--- a scalar and a vector two vectors two scalars .
(e) a · b + c has ---Select--- meaning no meaning because it is the sum of ---Select--- a scalar and a vector two vectors two scalars .
(f) |a| · (b + c) has ---Select--- meaning no meaning because it is the dot product of ---Select--- a scalar and a vector two vectors two scalars .

Answer 1

The expressions (a) (a · b) · c, (b) (a · b)c, (c) |a|(b · c), (d) a · (b + c), (e) a · b + c, and (f) |a| · (b + c) are all meaningful because they involve the dot product and scalar multiplication of vectors, resulting in scalar quantities.

Step-by-step explanation:

The expressions (a) (a · b) · c, (b) (a · b)c, (c) |a|(b · c), (d) a · (b + c), (e) a · b + c, and (f) |a| · (b + c) are all meaningful.These expressions involve the dot product and scalar multiplication of vectors, which result in scalar quantities.The dot product of two vectors is meaningful when it involves two vectors. The dot product of a scalar and a vector or two scalars is meaningless. Scalar multiplication of a scalar and a vector is meaningful. When the dot product is multiplied by the magnitude of another vector, it is also meaningful.

(f) |a| · (b + c) has no meaning because it incorrectly implies the dot product of a scalar and a vector, which is not defined in vector algebra.Each of these expressions must be scrutinized within the context of vector and scalar products to determine their validity. The key is to remember that scalar products involve two vectors and produce a scalar, whereas vector products involve two vectors to produce another vector, and scalars can multiply vectors but cannot be involved in dot products with vectors.