Chapter 7.5, Problem 59E

(a).

To determine

To calculate: the sum of two vectors using matrix .

Answer to Problem 59E

The addition of two vectors that is $\underset{\_}{u+v=\langle 4,7\rangle }$ .

Explanation of Solution

Given information:

Two vectors which needs to be added are as follows.

  $u=\langle 1,5\rangle$ and $v=\langle 3,2\rangle$ ..

Formula used:

  $u+v=\langle u_1,u_2\rangle +\langle v_1,v_2\rangle =\langle u_1+v_1,u_2+v_2\rangle$

Where vector $u=\langle u_1,u_2\rangle$ and $v=\langle v_1,v_2\rangle$

Calculation:

Two vectors are as follows.

  $u=\langle 1,5\rangle$ and $v=\langle 3,2\rangle$ .

Calculating $u+v$ using $u+v=\langle u_1,u_2\rangle +\langle v_1,v_2\rangle =\langle u_1+v_1,u_2+v_2\rangle$

  $u+v=\langle 1,5\rangle +\langle 3,2\rangle$

  $=\langle 1+3,5+2\rangle$

  $=\langle 4,7\rangle$

Hence the addition of two vectors that is $\underset{\_}{u+v=\langle 4,7\rangle }$ .

(b).

To determine

To calculate: the subtraction of two vectors using matrix .

Answer to Problem 59E

The subtraction of two vectors that is $\underset{\_}{u-v=\langle -2,3\rangle }$ .

Explanation of Solution

Given information:

Two vectors which needs to be subtracted are as follows.

  $u=\langle 1,5\rangle$ and $v=\langle 3,2\rangle$ ..

Formula used:

  $u-v=\langle u_1,u_2\rangle -\langle v_1,v_2\rangle =\langle u_1-v_1,u_2-v_2\rangle$

Where vector $u=\langle u_1,u_2\rangle$ and $v=\langle v_1,v_2\rangle$

Calculation:

Two vectors are as follows.

  $u=\langle 1,5\rangle$ and $v=\langle 3,2\rangle$ .

Calculating $u-v$ using $u-v=\langle u_1,u_2\rangle -\langle v_1,v_2\rangle =\langle u_1-v_1,u_2-v_2\rangle$

  $u-v=\langle 1,5\rangle -\langle 3,2\rangle$

  $=\langle 1-3,5-2\rangle$

  $=\langle -2,3\rangle$

Hence the subtraction of two vectors that is $\underset{\_}{u-v=\langle -2,3\rangle }$ .

(c).

To determine

To perform: the vector operation using matrix .

Answer to Problem 59E

  $\underset{\_}{3v-u=\langle 8,1\rangle }$ .

Explanation of Solution

Given information:

1. Two vectors on which operation needs to be perform are as follows.

  $u=\langle 1,5\rangle$ and $v=\langle 3,2\rangle$ ..

2. Need to calculate $3v-u$ .

Formula used:

1. $ku=k\langle u_1,u_2\rangle =\langle k{u}_1,k{u}_2\rangle$

Where vector $u=\langle u_1,u_2\rangle$ and $k$ is scalar quantity.

2. $u-v=\langle u_1,u_2\rangle -\langle v_1,v_2\rangle =\langle u_1-v_1,u_2-v_2\rangle$

Where vector $u=\langle u_1,u_2\rangle$ and $v=\langle v_1,v_2\rangle$

Calculation:

Two vectors are as follows.

  $u=\langle 1,5\rangle$ and $v=\langle 3,2\rangle$ .

Calculating $3v-u$

  $3v-u=3\langle 3,2\rangle -\langle 1,5\rangle$

  $=\langle 9,6\rangle -\langle 1,5\rangle$

  $=\langle 9-1,6-5\rangle$

  $=\langle 8,1\rangle$

Hence $\underset{\_}{3v-u=\langle 8,1\rangle }$ .