Chapter 11, Problem 29RE

(a)

To determine

To find: Component form of the vector $v$

Answer to Problem 29RE

  $v=\langle -10,6,7\rangle$

Explanation of Solution

Given information: Initial point of vector $v$ is (7,-4,3) and terminal point of vector $v$ is (-3,2,10).

Calculation:

We know that if $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ are initial and terminal points of a vector $a$ respectively, then component form of vector

  $a$ is

  $a=\langle x_2-x_1,y_2-y_1,z_2-z_1\rangle$

Therefore,

  $\begin{array}{l}v=\langle -3-7,2+4,10-3\rangle \\ v=\langle -10,6,7\rangle \end{array}$

(b)

To determine

To find: The magnitude of vector $v$

Answer to Problem 29RE

  $\left|v\right|=\sqrt{185}$

Explanation of Solution

Given information: Initial point of vector $v$ is (7,-4,3) and terminal point of vector $v$ is (-3,2,10).

Calculation:

Vector

  $v$ is

  $v=\langle -10,6,7\rangle$

Its magnitude is

  $\left|v\right|=\sqrt{{\left(-10\right)}^2+{\left(6\right)}^2+{\left(7\right)}^2}=\sqrt{185}$

(c)

To determine

To find: A unit vector in the direction of vector $v$

Answer to Problem 29RE

  $\frac{1}{\sqrt{185}}\langle -10,6,7\rangle$

Explanation of Solution

Given information: Initial point of vector $v$ is (7,-4,3) and terminal point of vector $v$ is (-3,2,10).

Calculation:

Vector

  $v$ is

  $v=\langle -10,6,7\rangle$

Its magnitude is $\left|v\right|=\sqrt{185}$

Now, unit vector in the direction of vector $v$ is

  $\widehat{v}=\frac{v}{\left|v\right|}=\frac{1}{\sqrt{185}}\langle -10,6,7\rangle$