Question

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

Vectors
ANSWERED
asked 2021-05-29
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 product of ?.
b) \((a\cdot b)c\)
\((a\cdot b)c\) has ? because it is a scalar multiple of ?.
c) \(|a|(b\cdot c)\)
\(|a|(b\cdot c)\) has ? because it is the product of ?.
d) \(a\cdot(b+c)\)
\(a\cdot(b+c)\) 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) \(|a|\cdot(b+c)\)
\(|a|\cdot(b+c)\) has ? because it is the dot product of ?.

Answers (1)

2021-05-30

Step 1
a) The expression \((a\cdot b)\cdot c\) 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 \((a\cdot b)c\) 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 \(|a|(b\cdot c)\) 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 \(a\cdot(b+c)\) 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 \(|a|\cdot(b+c)\) 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.

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