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3. State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (а) а (Ь x с) (d) а - (b- с) (b) ах (b-с) (e) (a - b) x (c - d) (c) a x (b x c) (f) (a x b) (c x d).

3. State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (а) а (Ь x с) (d) а - (b- с) (b) ах (b-с) (e) (a - b) x (c - d) (c) a x (b x c) (f) (a x b) (c x d).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Question 3: Expression Analysis State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. #### Expressions: **(a)** \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) **(b)** \( \mathbf{a} \times (\mathbf{b} \cdot \mathbf{c}) \) **(c)** \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) **(d)** \( \mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c}) \) **(e)** \( (\mathbf{a} \cdot \mathbf{b}) \times (\mathbf{c} \cdot \mathbf{d}) \) **(f)** \( (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) \) #### Detailed Explanation: - **Expression (a)** \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \): - **Meaningful**: Yes. - **Type**: Scalar. - **Explanation**: The cross product \( \mathbf{b} \times \mathbf{c} \) results in a vector. The dot product of vector \( \mathbf{a} \) with this vector is a scalar. - **Expression (b)** \( \mathbf{a} \times (\mathbf{b} \cdot \mathbf{c}) \): - **Meaningful**: No. - **Explanation**: The dot product \( \mathbf{b} \cdot \mathbf{c} \) results in a scalar. The cross product of vector \( \mathbf{a} \) with a scalar is not defined. - **Expression (c)** \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \): - **Meaningful**: Yes. - **Type**: Vector. - **Explanation**: The cross product \( \mathbf{b} \times \mathbf{c} \) results in a vector. The cross product of vector \( \mathbf{a} \) with this vector is a vector
Transcribed Image Text:### Question 3: Expression Analysis State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. #### Expressions: **(a)** \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) **(b)** \( \mathbf{a} \times (\mathbf{b} \cdot \mathbf{c}) \) **(c)** \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) **(d)** \( \mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c}) \) **(e)** \( (\mathbf{a} \cdot \mathbf{b}) \times (\mathbf{c} \cdot \mathbf{d}) \) **(f)** \( (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) \) #### Detailed Explanation: - **Expression (a)** \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \): - **Meaningful**: Yes. - **Type**: Scalar. - **Explanation**: The cross product \( \mathbf{b} \times \mathbf{c} \) results in a vector. The dot product of vector \( \mathbf{a} \) with this vector is a scalar. - **Expression (b)** \( \mathbf{a} \times (\mathbf{b} \cdot \mathbf{c}) \): - **Meaningful**: No. - **Explanation**: The dot product \( \mathbf{b} \cdot \mathbf{c} \) results in a scalar. The cross product of vector \( \mathbf{a} \) with a scalar is not defined. - **Expression (c)** \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \): - **Meaningful**: Yes. - **Type**: Vector. - **Explanation**: The cross product \( \mathbf{b} \times \mathbf{c} \) results in a vector. The cross product of vector \( \mathbf{a} \) with this vector is a vector
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