Let u = [ 1, 1, 4 ] and v = [ 3, 2, 4 ].
(a) Find a vector n = [ x, y, z ] that is orthogonal to u and orthogonal to v. Hint: Use the dot product to create a system of linear equations, then use Gauss-Jordan to solve that SLE.
(b) Use your answer in (a) to find the normal and general forms of the plane x = [3, 1, 1]+su+tv.
Solution
a) the dot product is multiplying and adding the components of each vector. for a vector to be orthogonal to another vector the dot product has to equal zero. (1x+1y+4z)=0 any point on it is orthogonal (3x+2y+4z)=0 1x=3x 1y=2y 4z=4z so a vector orthogonal would be [1,1,2], [1,1,3], any value for the z will work.
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Let u - - 1- 1- 4 - and v - - 3- 2- 4 -- (a) Find a vector n - - x- y-.docx
1. Let u = [ 1, 1, 4 ] and v = [ 3, 2, 4 ].
(a) Find a vector n = [ x, y, z ] that is orthogonal to u and orthogonal to v. Hint: Use the dot
product to create a system of linear equations, then use Gauss-Jordan to solve that SLE.
(b) Use your answer in (a) to find the normal and general forms of the plane x = [3, 1, 1]+su+tv.
Solution
a) the dot product is multiplying and adding the components of each vector. for a vector to be
orthogonal to another vector the dot product has to equal zero. (1x+1y+4z)=0 any point on it is
orthogonal (3x+2y+4z)=0 1x=3x 1y=2y 4z=4z so a vector orthogonal would be [1,1,2], [1,1,3],
any value for the z will work.
![Let u = [ 1, 1, 4 ] and v = [ 3, 2, 4 ].
(a) Find a vector n = [ x, y, z ] that is orthogonal to u and orthogonal to v. Hint: Use the dot
product to create a system of linear equations, then use Gauss-Jordan to solve that SLE.
(b) Use your answer in (a) to find the normal and general forms of the plane x = [3, 1, 1]+su+tv.
Solution
a) the dot product is multiplying and adding the components of each vector. for a vector to be
orthogonal to another vector the dot product has to equal zero. (1x+1y+4z)=0 any point on it is
orthogonal (3x+2y+4z)=0 1x=3x 1y=2y 4z=4z so a vector orthogonal would be [1,1,2], [1,1,3],
any value for the z will work.](https://image.slidesharecdn.com/letu-1-1-4-andv-3-2-4-afindavectorn-x-y-230128215750-d4075010/85/Let-u-1-1-4-and-v-3-2-4-a-Find-a-vector-n-x-y-docx-1-320.jpg)
