Answer to Question #276067 in Linear Algebra for Nikhil

Question #276067

Complete { (2, 0, 3)} to form an orthogonal



basis of R³

1
Expert's answer
2021-12-06T17:15:37-0500

Let us complete {(2,0,3)}\{ (2, 0, 3)\} to form an orthogonal basis. Consider the vector (3,0,2).(3,0,-2). Since 23+00+3(2)=0,2\cdot 3+0\cdot0+3\cdot(-2)=0, we conclude that the inner product of (2,0,3)(2, 0, 3) and (3,0,2)(3,0,-2) is zero, and hence the vectors (2,0,3)(2, 0, 3) and (3,0,2)(3,0,-2) are orthogonal. Further, consider the vector (0,1,0)(0,1,0). It follows that 20+01+30=02\cdot 0+0\cdot 1+3\cdot 0=0 and 30+0120=0,3\cdot 0+0\cdot 1-2\cdot 0=0, and hence the vector (0,1,0)(0,1,0) is orthogonal to the vectors (2,0,3)(2, 0, 3) and (3,0,2).(3,0,-2). Therefore, {(2,0,3),(3,0,2),(0,1,0)}\{ (2, 0, 3),(3,0,-2),(0,1,0)\} is an orthogonal basis of R3.\R^3.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment