Question

let ( mathbf{v} = -3mathbf{i} + 5mathbf{j} ) and ( mathbf{w} = -mathbf{i} - 4mathbf{j} ). find ( 7mathbf{v} - 5mathbf{w} ).( 7mathbf{v} - 5mathbf{w} = square ) (simplify your answer. type your answer in terms of ( mathbf{i} ) and ( mathbf{j} ).)

Explanation:

Step1: Calculate 7v

$7\mathbf{v} = 7(-3\mathbf{i} + 5\mathbf{j}) = -21\mathbf{i} + 35\mathbf{j}$

Step2: Calculate 5w

$5\mathbf{w} = 5(-\mathbf{i} - 4\mathbf{j}) = -5\mathbf{i} - 20\mathbf{j}$

Step3: Subtract 5w from 7v

$7\mathbf{v} - 5\mathbf{w} = (-21\mathbf{i} + 35\mathbf{j}) - (-5\mathbf{i} - 20\mathbf{j})$
$= -21\mathbf{i} + 35\mathbf{j} + 5\mathbf{i} + 20\mathbf{j}$

Step4: Combine like terms

$= (-21 + 5)\mathbf{i} + (35 + 20)\mathbf{j} = -16\mathbf{i} + 55\mathbf{j}$

Answer:

$-16\mathbf{i} + 55\mathbf{j}$