LinAlg4.2.2

From Example Problems
Jump to: navigation, search

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