silemus.
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  1. 23×33\times 3 determinant expansion
  2. 10Eigenvalues of a 3×33\times 3 matrix
  3. 12Eigenvector for eigenvalue 3
  4. 14Inverse of a 2×22\times 2 matrix
  5. 17Solving a 3×33\times 3 linear system
  6. 20System with no solution
  7. 21Singular matrix, det=0
  8. 22Eigenvalue zero of a matrix
  9. 27Null space of a matrix
  10. 37Volume via scalar triple product
  11. 43Characteristic polynomial, 2×22\times 2
  12. 47Orthogonal matrix check
  13. 48Gram-Schmidt second vector
  14. 49Least-squares line fit
  15. 50Coordinates in a new basis
  16. 51Distance point to plane

51Distance point to plane

Find the distance from the point (1,2,3)\displaystyle (1, 2, 3) to the plane x+2y+2z=0\displaystyle x + 2y + 2z = 0. Give the answer to 3 decimal places.

Use the point-to-plane formula: distance equals the plane's left side evaluated at the point, divided by the length of the normal vector (1,2,2). That's |1+4+6|/√9 = 11/3 ≈ 3.667.

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