Chapter 10, Problem 2RE

a)

To determine

To find: The component form of the vector $u+v$ .

Answer to Problem 2RE

The component form of the vector $u+v$ is $〈-1,-1〉$ .

Explanation of Solution

Given information:

The given vectors are $u=〈-3,4〉$ and $v=〈2,-5〉$ .

Formula used:

Vector Addition:

Let $u=〈u_1,u_2〉$ and $v=〈v_1,v_2〉$ be vectors, then the sum/ resultant of the vectors $u$ and $v$ is given by:

  $\begin{array}{c}u+v=〈u_1,u_2〉+〈v_1,v_2〉\\ =〈u_1+v_1,u_2+v_2〉\end{array}$

Component form of a vector:

A two-dimensional vector $v$ is an ordered pair of real numbers, denoted in component form as

  $〈a,b〉$ .The numbers $a$ and $b$ are the components of the vector $v$ .

Calculation:

Substitute $u=〈-3,4〉$ and $v=〈2,-5〉$ , and apply the formula for vector addition to find $u+v$ :

  $\begin{array}{c}u+v=〈-3,4〉+〈2,-5〉\\ =〈-3+2,4+\left(-5\right)〉\\ =〈-1,-1〉\end{array}$

According to the Component form of vectors,

The component form of the vector $u+v$ is $〈-1,-1〉$ .

b)

To determine

To find:The magnitude of the vector $u+v$ .

Answer to Problem 2RE

The magnitude of the vector $u+v$ is $1.414$ .

Explanation of Solution

Given information:The given vectors are $u=〈-3,4〉$ and $v=〈2,-5〉$ .

Formula used:

Magnitude of a vector:

The magnitude of the vector $〈a,b〉$ is the non-negative real number:

  $\left|〈a,b〉\right|=\sqrt{a^2+b^2}$ .

Vector Addition:

Let $u=〈u_1,u_2〉$ and $v=〈v_1,v_2〉$ be vectors, then the sum/ resultant of the vectors $u$ and $v$ is given by:

  $\begin{array}{c}u+v=〈u_1,u_2〉+〈v_1,v_2〉\\ =〈u_1+v_1,u_2+v_2〉\end{array}$

Calculation:

Substitute $u=〈-3,4〉$ and $v=〈2,-5〉$ , and apply the formula for vector addition to find $u+v$ :

  $\begin{array}{c}u+v=〈-3,4〉+〈2,-5〉\\ =〈-3+2,4+\left(-5\right)〉\\ =〈-1,-1〉\end{array}$

Substitute $u+v=〈-1,-1〉$ and apply the formula for magnitude of vectors to find the value of $\left|u+v\right|$ :

  $\begin{array}{c}\left|u+v\right|=\left|〈-1,-1〉\right|\\ =\sqrt{{\left(-1\right)}^2+{\left(-1\right)}^2}\\ =\sqrt{1+1}\\ =\sqrt{2}\\ \approx 1.414\end{array}$

Thus, the magnitude of the vector $u+v$ is $1.414$ .