Vectors: Combinations defined or not 1

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Question statement

Given the vectors $\boldsymbol{A},\;\;\boldsymbol{B}$ and $\boldsymbol{C}$ in $3$ dimensional space, state whether the following quantities are scalars, vectors or undefined.

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$\boldsymbol{(A\cdot B)\cdot C}$
$\boldsymbol{(A\cdot B)C}$
$\boldsymbol{(A\cdot B)\times C}$
$\boldsymbol{(A\times B)\times C}$
$\boldsymbol{(A\times B)\cdot C}$

Scalar
Vector
Undefined

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	Scalar	Vector	Undefined
$\boldsymbol{(A\cdot B)\cdot C}$
$\boldsymbol{(A\cdot B)C}$
$\boldsymbol{(A\cdot B)\times C}$
$\boldsymbol{(A\times B)\times C}$
$\boldsymbol{(A\times B)\cdot C}$

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1. $\boldsymbol{(A\cdot B)\cdot C}$ is undefined as $\boldsymbol{A\cdot B}$ is a scalar and we cannot take the inner product of a scalar with the vector $\boldsymbol{C}$ .

2. $\boldsymbol{(A\cdot B)C}$ is a vector and is a multiple of $\boldsymbol{C}$ as $\boldsymbol{A \cdot B}$ is a scalar.

3. $\boldsymbol{(A\cdot B)\times C}$ is undefined as $\boldsymbol{A\cdot B}$ is a scalar and the cross product is only defined between vectors.

4. $\boldsymbol{(A\times B)\times C}$ is a vector as $\boldsymbol{A \times B}$ and $\boldsymbol{C}$ are vectors and the cross product between vectors produces a vector.

5. $\boldsymbol{(A\times B)\cdot C}$ is a scalar as $\boldsymbol{A \times B}$ and $\boldsymbol{C}$ are vectors and the inner or dot product is between vectors and produces a scalar.

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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