Question-7

Let \(v_{1}=(1,0,0),v_{2}=(0,1,0)\) and \(v_{3}=(0,0,1)\) be three vectors and let \(a,b,c\) be three real numbers. Express the following vectors in terms of \(v_{1},v_{2}\) and \(v_{3}\).

• \((a,b,c)\)

• \((a,0,c)\)

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Notice the following:

\[ \begin{aligned} a\cdot v_{1} & =a\cdot(1,0,0)=(a,0,0)\\ b\cdot v_{2} & =b\cdot(0,1,0)=(0,b,0)\\ c\cdot v_{3} & =c\cdot(0,0,1)=(0,0,c) \end{aligned} \]

Therefore, we have:

\[ \begin{aligned} (a,b,c) & =av_{1}+bv_{2}+cv_{3}\\ (a,0,c) & =av_{1}+cv_{3} \end{aligned} \]

We have expressed the vectors \((a,b,c)\) and \((a,0,c)\) as a linear combination of the vectors \(v_{1},v_{2}\) and \(v_{3}\). This is a term that you will see throughout the course.