LinAlg4.2.2

From Exampleproblems

Find a vector $\bar{d}\,$ which is perpendicular to both $\bar{a}=4\bar{i}+5\bar{j}-\bar{k},\bar{b}=\bar{i}-4\bar{j}+5\bar{k}\,$ and $\bar{d}\cdot \bar{c}=21\,$ where $\bar{c}=3\bar{i}+\bar{j}-\bar{k}\,$

Let $\bar{d}=l\bar{i}+m\bar{j}+n\bar{k}\,$ where l,m,n are scalars.

Given $\bar{a}=4\bar{i}+5\bar{j}-\bar{k},\bar{b}=\bar{i}-4\bar{j}+5\bar{k},\bar{c}=3\bar{i}+\bar{j}-\bar{k}\,$

Also given $\bar{d}\cdot\bar{c}=21,\bar{d}\cdot\bar{a}=0,\bar{d}\cdot\bar{b}=0\,$ implies

$l(4)+m(5)+n(-1)=0,l(1)+m(-4)+n(5)=0,l(3)+m(1)+n(-1)=21\,$

$4l+5m-n=0,l-4m+5n=0,3l+m-n=21\,$

From the first two equations,we have $\frac{l}{1}=\frac{m}{-1}=\frac{n}{-1}=x\,$ (say)

$3(x)-x+x=21,x=7\,$

Therefore,the required vector is $7\bar{i}-7\bar{j}-7\bar{k}\,$

Main Page:Linear Algebra

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