Vectors 3 (dot and cross)

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Violeta CIT 7 years, 6 months ago

Created this as a copy of Dot and cross product combinations.

Name	Status	Author	Last Modified
Dot and cross product combinations	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Vectors 3 (dot and cross)	draft	Violeta CIT	09/03/2023 16:32

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Question statement

Given the vectors $\boldsymbol{v}$ , $\boldsymbol{w}$ , $\boldsymbol{u}$ in $\mathbb{R}^3$ , state whether the following quantities are scalars (real numbers), vectors (elements of $\mathbb{R}^3$ ) or undefined.

In this question, the symbol $\cdot$ denotes the inner product and $\times$ always denotes the cross product.

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			Name	Type	Generated Value

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$\boldsymbol{(v\cdot w)\cdot u}$
$\boldsymbol{(v\cdot w)u}$
$\boldsymbol{(v\cdot w)\times u}$
$\boldsymbol{(v\times w)\times u}$
$\boldsymbol{(v\times w)\cdot u}$

Scalar
Vector
Undefined

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	Scalar	Vector	Undefined
$\boldsymbol{(v\cdot w)\cdot u}$
$\boldsymbol{(v\cdot w)u}$
$\boldsymbol{(v\cdot w)\times u}$
$\boldsymbol{(v\times w)\times u}$
$\boldsymbol{(v\times w)\cdot u}$

Advice

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

2. $\boldsymbol{(v\cdot w) u}$ is a vector and is a scalar multiple of $\boldsymbol{u}$ as $\boldsymbol{v \cdot w}$ is a scalar.

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

4. $\boldsymbol{(v\times w)\times u}$ is a vector as $\boldsymbol{v \times w}$ and $\boldsymbol{u}$ are vectors and the cross product between vectors produces a vector.

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

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Violeta CIT 7 years, 6 months ago

Error in variable testing condition

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