Chapter 6.3, Problem 117E

(a)

To determine

The geometric descriptions of a vector addition

Answer to Problem 117E

let $u=(u_1,u_2)$

and $v=(v_1,v_2)$

be vectors. Then the sum of u and v is the vector

  $u+v=(u_1+v_1,u_2+v_2)$

Explanation of Solution

Concept used: let $u=(u_1,u_2)$

and $v=(v_1,v_2)$

be vectors. Then the sum of u and v is the vector

  $u+v=(u_1+v_1,u_2+v_2)$

Calculation:

To add two vectors geometrically position them so that the initial point of one coincides with the terminal points of the other

The sum $u+v$ is formed by joining the initial point of the second vector $v$ with the terminal point of the first vector $u$ . as shown in figure

This technique is called the parallelogram law for vector addition because the vector $u+v$ , often called the resultant of vector addition, is the diagonal of a parallelogram having $u$ and $v$ as its adjacent sides

(b)

To determine

The geometric descriptions of a vector scalar multipication

Answer to Problem 117E

  $u=(u_1,u_2)$ and let k be a scalar then the scalar multiple of $k$ times $u$ is the vector

  $\begin{array}{l}ku=k(u_1,u_2)\\ ku=(k{u}_1,k{u}_2)\end{array}$

Explanation of Solution

Concept used: let $u=(u_1,u_2)$

and let $k$ be a scalar then the scalar multiple of $k$ times $u$ is the vector

  $\begin{array}{l}ku=k(u_1,u_2)\\ ku=(k{u}_1,k{u}_2)\end{array}$

Calculation:

The product of a vector v and a scalar $k$ is the vector that is | $k$ | times as long $v$ . if $k$ is positive, $kv$ has the same direction as $v$ , and if $k$ is negative, $kv$ has the direction opposite that of $v$

As shown in the figure