Question

Given that $\overrightarrow{\mathbf{v}}\mathbf{=}\mathbf{<}{\mathbf{v}}_{\mathbf{x}}\mathbf{,}{\mathbf{v}}_{\mathbf{y}}\mathbf{,}{\mathbf{v}}_{\mathbf{z}}\mathbf{>}$and $\hat{\mathbf{r}}\mathbf{=}\frac{<x,y,z>}{\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}}}$, write out role="math" localid="1655386437949" $\overrightarrow{\mathbf{v}}\mathbf{\times }\hat{\mathbf{r}}$ as a vector.

Short answer

The $\overrightarrow{\mathrm{v}}\times \hat{\mathrm{r}}$ as a vector is $\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2}}\left[\left({\mathrm{v}}_{\mathrm{y}}\mathrm{z}-{\mathrm{v}}_{\mathrm{z}}\mathrm{y}\right)\hat{\mathrm{i}}-\left({\mathrm{v}}_{\mathrm{x}}\mathrm{z}-{\mathrm{v}}_{\mathrm{z}}\mathrm{x}\right)\hat{\mathrm{j}}-\left({\mathrm{v}}_{\mathrm{x}}\mathrm{y}-{\mathrm{v}}_{\mathrm{y}}\mathrm{x}\right)\right]$.

Step-by-step solution

Identification of the given data

The given data can be listed below as,

• The vector $\overrightarrow{\mathrm{v}}$ is, .

• The vector $\overrightarrow{\mathrm{r}}$is, .

Significance of vector product (cross product)

Whenever the vector product of two particular vector quantities is performed, then it results in the vector quantity. The vector quantity denotes the magnitude along with its direction also.

Determination the vector product of two vectors

The vector product of two vectors can be calculated as,

Thus, the $\overrightarrow{v}\times \hat{r}$ as a vector is$\overrightarrow{\mathrm{v}}\times \hat{\mathrm{r}}=\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2}}\left[\left({\mathrm{v}}_{\mathrm{y}}\mathrm{z}-{\mathrm{v}}_{\mathrm{z}}\mathrm{y}\right)\hat{\mathrm{i}}-\left({\mathrm{v}}_{\mathrm{x}}\mathrm{z}-{\mathrm{v}}_{\mathrm{z}}\mathrm{x}\right)\hat{\mathrm{j}}+\left({\mathrm{v}}_{\mathrm{x}}\mathrm{y}-{\mathrm{v}}_{\mathrm{y}}\mathrm{x}\right)\right]$

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