Chapter 17: 1Q (page 673)

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

Short Answer

Expert verified

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

Step by step solution

Identification of the given data

The given data can be listed below as,

The vector $\vec{v}$ is, $\vec{v} = <v_{x}, v_{y}, v_{z}>$ .

The vector $\vec{r}$ is, $\hat{r} = \frac{<x, y, z>}{\sqrt{x^{2} + y^{2} + z^{2}}}$ .

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,

$\vec{v} \times \hat{r} = <v_{x}, v_{y}, v_{z}> \times \frac{(x, y, z)}{\sqrt{x^{2} + y^{2} + z^{2}}} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ v_{x} & v_{y} & v_{z} \\ \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}} & \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}} & \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \end{matrix}| = \{\begin{matrix} \hat{i} [\frac{v_{y} z}{\sqrt{x^{2} + y^{2} + z^{2}}} - \frac{v_{z} y}{\sqrt{x^{2} + y^{2} + z^{2}}}] \\ - \hat{j} [\frac{v_{x} z}{\sqrt{x^{2} + y^{2} + z^{2}}} - \frac{v_{z} x}{\sqrt{x^{2} + y^{2} + z^{2}}}] \\ + \hat{k} [\frac{v_{x} y}{\sqrt{x^{2} + y^{2} + z^{2}}} - \frac{v_{y} x}{\sqrt{x^{2} + y^{2} + z^{2}}}] \end{matrix}\} = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}} [(v_{y} z - v_{z} y) \hat{i} - (v_{x} z - v_{z} x) \hat{j} (v_{x} y - v_{y} x)]$

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

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

Short Answer

Step by step solution

Identification of the given data

Significance of vector product (cross product)

Determination the vector product of two vectors

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Given that v→=<vx,vy,vz>and r^=<x,y,z>x2+y2+z2, write out role="math" localid="1655386437949" v→×r^ as a vector.

Short Answer

Step by step solution

Identification of the given data

Significance of vector product (cross product)

Determination the vector product of two vectors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Magnetism

Electric Charge Field and Potential

Classical Mechanics

Kinematics Physics

Electricity

Study anywhere. Anytime. Across all devices.

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