Vectors 3 Question 20
20. Which of the following expressions are meaningfull operations?
(1998, 2M)
(a) $\overrightarrow{\mathbf{u}} \cdot(\overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{w}})$
(b) $(\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}}) \cdot \overrightarrow{\mathbf{w}}$
(c) $(\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}}) \overrightarrow{\mathbf{w}}$
(d) $\overrightarrow{\mathbf{u}} \times(\overrightarrow{\mathbf{v}} \cdot \overrightarrow{\mathbf{w}})$
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Answer:
Correct Answer: 20. (a, c)
Solution:
- (a) $\overrightarrow{\mathbf{u}} \cdot(\overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{w}})$ is a meaningful operation.
Therefore, (a) is the answer.
(b) $\overrightarrow{\mathbf{u}} \cdot(\overrightarrow{\mathbf{v}} \cdot \overrightarrow{\mathbf{w}})$ is not meaningful, since $\overrightarrow{\mathbf{v}} \cdot \overrightarrow{\mathbf{w}}$ is a scalar quantity and for dot product both quantities should be vector.
Therefore, (b) is not the answer.
(c) $(\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}}) \overrightarrow{\mathbf{w}}$ is meaningful, since it is a simple multiplication of vector and scalar quantity. Therefore, (c) is the answer.
(d) $\overrightarrow{\mathbf{u}} \times(\overrightarrow{\mathbf{v}} \cdot \overrightarrow{\mathbf{w}})$ is not meaningful, since $\overrightarrow{\mathbf{v}} \cdot \overrightarrow{\mathbf{w}}$ is a scalar quantity and for cross product, both quantities should be vector.
Therefore, (d) is not the answer.
