Chapter 6, Problem 2RE

Solution

To find: The value of the expression $2u-3w$ where $u=〈2,-1〉$ , $v=〈4,2〉$ and $w=〈1,-3〉$

The value of the expression $2u-3w$ where $u=〈2,-1〉$ , $v=〈4,2〉$ and $w=〈1,-3〉$ is $〈1,7〉$ .

Given information:

The given vectors are :

  $u=〈2,-1〉$

  $v=〈4,2〉$

  $w=〈1,-3〉$

Formula used:

Vector addition:

Let $u=〈u_1,u_2〉$ and $v=〈v_1,v_2〉$ be two vectors. The sum (or resultant ) of the vectors $u$ and $v$ is the vector

  $u+v=〈u_1+v_1,u_2+v_2〉$

Scalar Multiplication:

Let $u=〈u_1,u_2〉$ and $v=〈v_1,v_2〉$ be two vectors and let $k$ be a real number (scalar).

The product of the scalar $k$ and the vector $u$ is

  $\begin{array}{c}ku=k〈u_1,u_2〉\\ =〈k{u}_1,k{u}_2〉\end{array}$

Calculation:

Substitute $〈2,-1〉$ for $u$ and $〈1,-3〉$ for $w$ to calculate $2u$ .

  $\begin{array}{c}2u=2〈2,-1〉\\ =〈2\times 2,2\times (-1)〉\\ =〈4,-2〉\end{array}$

Substitute $〈2,-1〉$ for $u$ and $〈1,-3〉$ for $w$ to calculate $3w$ .

  $\begin{array}{c}3w=3〈1,-3〉\\ =〈3\times 1,3\times (-3)〉\\ =〈3,-9〉\end{array}$

Substitute $〈2,-1〉$ for $u$ and $〈1,-3〉$ for $w$ to determine the value of the expression

  $2u-3w$ .

  $\begin{array}{c}2u+(-3)w=2u-3w\\ =〈4,-2〉-〈3,-9〉\\ =〈4-3,-2-(-9)〉\\ =〈1,7〉\end{array}$

Hence, the value of the expression $2u-3w$ where $u=〈2,-1〉$ , $v=〈4,2〉$ and $w=〈1,-3〉$ is $〈1,7〉$ .