Q. 17 - Calculus

Question

Let $v_0=(a, b, c)$ and let $v=(x, y, z)$ Describe the sets of points in $R^3$ satisfying the following properties:

$$\begin{gathered} \left(a\right). \left|v\right|=4 \\ \left(b\right). \left|v\right|\le 4 \\ \left(c\right). \left|v-v_0\right|=4 \end{gathered}$$

Step-by-Step Solution

Answer

(a) The set of points are $(0, 0, 4), (0, 0, -4), (0, 4, 0), (0, -4, 0), (4, 0, 0) \mathrm{and} (-4, 0, 0) \mathrm{etc}.$

(b)  The set of values of $(x, y, z)$ , which satisfy the relation $x^2+y^2+z^2\le 16$ will be the required set of points.

The set of points are, 

$(0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 1, 1), (0, 1, 2), (0, 1, 3), (0, 2, 1), (0, 2, 2), (0, 3, 1), (0, 0, 4), (0, 0, -4), (1, 2, 3) \mathrm{etc}.$

(c)  The set of values of $(x, y, z) \mathrm{and} (a, b, c)$, which satisfy the relation ${(x-a)}^2+{(y-b)}^2+{(z-c)}^2=16$ will be the required set of points.

The set of points $(x, y, z)$ are $(5, 1, 0), (4, 1, 0) \mathrm{and} (1, 5, 0) \mathrm{etc}.$

The set of points $(a, b, c)$ are $(1, 1, 0), (0, 1, 0) \mathrm{etc}$.

Step 1: Part (a)

Given that, $v=(x, y, z)$

$$\begin{gathered} \left|v\right|=\sqrt{x^2+y^2+z^2} \\ \sqrt{x^2+y^2+z^2}=4 \\ x^2+y^2+z^2=16 (\mathrm{Squaring} \mathrm{both} \mathrm{side}) \end{gathered}$$

When, $x=0, y=0$

$$\begin{gathered} z=\sqrt{16} \\ z=\pm 4 \end{gathered}$$

When $y=0, z=0$

$$\begin{gathered} x=\sqrt{16} \\ x=\pm 4 \end{gathered}$$

When, $x=0, z=0$

$$\begin{gathered} y=\sqrt{16} \\ y=\pm 4 \end{gathered}$$

Thus, the set of points are $(0, 0, 4), (0, 0, -4), (0, 4, 0), (0, -4, 0), (4, 0, 0) \mathrm{and} (-4, 0, 0) \mathrm{etc}.$

Step 1: Part (b)

Given that, $v=(x, y, z)$

$$\begin{gathered} \left|v\right|=\sqrt{x^2+y^2+z^2} \\ \sqrt{x^2+y^2+z^2}\le 4 \\ x^2+y^2+z^2\le 16 (\mathrm{Squaring} \mathrm{both} \mathrm{side}) \end{gathered}$$

The set of values of $(x, y, z)$, which satisfy the relation $x^2+y^2+z^2\le 16$ will be the required set of points.

The set of points are, $(0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 1, 1), (0, 1, 2), (0, 1, 3), (0, 2, 1), (0, 2, 2), (0, 3, 1), (0, 0, 4), (0, 0, -4), (1, 2, 3) \mathrm{etc}.$

Step 1: Part (c)

Given that,

$$\begin{gathered} v=(x, y, z) \mathrm{and} v_0=(a, b, c) \\ v-v_0=\left\{(x-a), (y-b), (z-c)\right\} \\ \left|v-v_0\right|=\sqrt{{(x-a)}^2+{(y-b)}^2+{\left(z-c\right)}^2} \\ \sqrt{{(x-a)}^2+{(y-b)}^2+{(z-c)}^2}=4 \left(\mathrm{Given}\right) \\ { (x-a)}^2+{(y-b)}^2+{(z-c)}^2=16 \left(\mathrm{Squaring} \mathrm{both} \mathrm{side}\right) \end{gathered}$$

The set of values of $(x, y, z) \mathrm{and} (a, b, c)$, which satisfy the relation ${(x-a)}^2+{(y-b)}^2+{(z-c)}^2=16$ will be the required set of points.

The set of points $(x, y, z)$ are $(5, 1, 0), (4, 1, 0) \mathrm{and} (1, 5, 0) \mathrm{etc}.$

The set of points $(a, b, c)$ are $(1, 1, 0), (0, 1, 0) \mathrm{etc}$.