Chapter 13.4, Problem 32WE

a.

To determine

To prove: that $k\left[\left(a,b\right)+\left(c,d\right)\right]=k\left(a,b\right)+k\left(c,d\right)$ .

Explanation of Solution

Given information: The expression to prove is:

  $k\left[\left(a,b\right)+\left(c,d\right)\right]=k\left(a,b\right)+k\left(c,d\right)$ .

Calculation:

Apply second definition of vector addition.

  $k\left[\left(a,b\right)+\left(c,d\right)\right]=k\left[\left(a+c\right),\left(b+d\right)\right]$

Apply first definition of vector addition.

  $k\left[\left(a+c\right),\left(b+d\right)\right]=\left[k\left(a+c\right),k\left(b+d\right)\right]$

Further simplify the above expression.

  $\left[k\left(a+c\right),k\left(b+d\right)\right]=k\left(a,b\right)+k\left(c,d\right)$

Therefore, $k\left[\left(a,b\right)+\left(c,d\right)\right]=k\left(a,b\right)+k\left(c,d\right)$ is proved.

b.

To determine

To draw: diagram to represent the expression $k\left[\left(a,b\right)+\left(c,d\right)\right]=k\left(a,b\right)+k\left(c,d\right)$ .

Explanation of Solution

Given information:

The expression to represent using vector is:

  $k\left[\left(a,b\right)+\left(c,d\right)\right]=k\left(a,b\right)+k\left(c,d\right)$ .

Calculation:

Let $P=(a,b)andQ=\left(c,d\right)$ . $k\left[\left(a,b\right)+\left(c,d\right)\right]=k\left(a,b\right)+k\left(c,d\right)$

Represent expression using vector.