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Which of the following expressions are...

Which of the following expressions are meaningful? ` vec u.( vec vxx vec w)` b. `( vec u. vec v). vec w` c. `( vec u. vec v). vec w` d. ` vec uxx( vec v. vec w)`

A

`vecu.(vecvxx vecw)`

B

`(vecu.vecv).vecw`

C

`(vecu.vecv)vecw`

D

`vecu xx (vecv . Vecw)`

Text Solution

AI Generated Solution

The correct Answer is:
To determine which of the given vector expressions are meaningful, we need to analyze each option based on the rules of vector operations, specifically the dot product and the cross product. ### Step-by-Step Solution: 1. **Option A: \( \vec{u} \cdot (\vec{v} \times \vec{w}) \)** - The expression involves the cross product \( \vec{v} \times \vec{w} \), which results in a vector (let's call it \( \vec{k} \)). - Then, we take the dot product of \( \vec{u} \) with \( \vec{k} \): \( \vec{u} \cdot \vec{k} \). - The dot product of two vectors results in a scalar. - Therefore, this expression is meaningful. 2. **Option B: \( (\vec{u} \cdot \vec{v}) \cdot \vec{w} \)** - The first part \( \vec{u} \cdot \vec{v} \) results in a scalar (let's call it \( s \)). - The second part involves taking the dot product of a scalar \( s \) with a vector \( \vec{w} \). - The dot product is defined only between two vectors, not between a scalar and a vector. - Therefore, this expression is not meaningful. 3. **Option C: \( (\vec{u} \cdot \vec{v}) \cdot \vec{w} \)** - This option is identical to Option B. - As previously analyzed, \( \vec{u} \cdot \vec{v} \) results in a scalar, and taking the dot product of a scalar with a vector is not defined. - Therefore, this expression is not meaningful. 4. **Option D: \( \vec{u} \times (\vec{v} \cdot \vec{w}) \)** - The expression involves the dot product \( \vec{v} \cdot \vec{w} \), which results in a scalar (let's call it \( t \)). - The next part involves taking the cross product of \( \vec{u} \) with a scalar \( t \). - The cross product is defined only between two vectors, so a scalar cannot be crossed with a vector. - Therefore, this expression is not meaningful. ### Summary of Meaningfulness: - **Option A**: Meaningful - **Option B**: Not Meaningful - **Option C**: Not Meaningful - **Option D**: Not Meaningful Thus, the only meaningful expression is **Option A**.
To determine which of the given vector expressions are meaningful, we need to analyze each option based on the rules of vector operations, specifically the dot product and the cross product. ### Step-by-Step Solution: 1. **Option A: \( \vec{u} \cdot (\vec{v} \times \vec{w}) \)** - The expression involves the cross product \( \vec{v} \times \vec{w} \), which results in a vector (let's call it \( \vec{k} \)). - Then, we take the dot product of \( \vec{u} \) with \( \vec{k} \): \( \vec{u} \cdot \vec{k} \). - The dot product of two vectors results in a scalar. ...
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