# \ #

# "Let:" \qquad A = i+2j, \quad B = j+2k; \quad \ \ L =2i+j+2k. #

# "Let:" \qquad P = "the plane formed by" \ \ A \ \ "and" \ \ B. #

# "Here's the idea:" #

- # N \ = \ A xx B \ \ "will be a vector normal to the plane" \ \ P.#
- # D \ = \ L xx N \ \ "will be a vector perpendicular to both" \ L \ "&" \ N #
- # "Because" \ D \ \ "is perpendicular to" \ N, "and" \ N \ "is normal to" \ P, #

# "we have" \ D \ \ "is in the plane" \ P. #
- # "So" \ D \ "will be both in the plane" \ P, "and perpendicular to" \ L, \ #

#"by (2)". #
- # "So, after normalizing" \ D \ "to a unit vector" \ \hat{D}, #

# \hat{D} \ "will have all the desired properties, and"#

# "will be our answer."#

# "Now we compute" \ D, \ "and then normalize it."#

# "Following the definitons of the vectors above, we have:" #

# \qquad \qquad \qquad \qquad \qquad D \ = \ L xx N \ = \ L xx (A xx B). #

# "Thus:" #

# D \ = \ L xx (A xx B)\ = \ L xx ( ( i+2j ) xx ( j+2k ) ) #

# \quad \quad = \ L xx | (i, j, k), (1, 2, 0), (0, 1, 2) | #

# \quad \quad = \ L xx ( |(2, 0), (1, 2)| i - |(1, 0), (0, 2)| j + |(1, 2), (0, 1)| k ) #

# \quad \quad = \ L xx ( 4 i - 2 j + k ) #

# \quad \quad = \ ( 2i + j +2k) xx ( 4 i - 2 j + k ) #

# \quad \quad = \ | (i, j, k), (2, 1, 2), (4, -2, 1) | #

# \quad \quad = \ ( |(1, 2), (-2, 1)| i - |(2, 2), (4, 1)| j + |(2, 1), (4, -2)| k ) #

# \quad \quad = \ ( 5 i + 6 j - 8k ). #

# "Hence:" \qquad \qquad \qquad \qquad \qquad \quad D = 5 i + 6 j - 8k. #

# "Recalling (4) above:" #

# \qquad \quad D \ "will be both in the plane" \ P, "and perpendicular to" \ L. #

# \qquad \qquad :. \quad \ D \ "is what we want, after normalizing it to"\ \hat{D}. #

# "Normalization of" \ D \ "to" \ \hat{D}:#

# "Hence:" \qquad \hat(D) = D/|| D || \ = \ ( 5 i + 6 j - 8k )/\sqrt{ 5^2 + 6^2 + (-8)^2 } #

# \qquad \qquad \qquad \qquad \qquad = \ ( 5 i + 6 j - 8k )/\sqrt{ 125 } \ = \ 1/{ 5 \sqrt{5} } ( 5 i + 6 j - 8k ). #

# "So we have the desired vector": " \quad \quad \hat{D} \ = \ 1/{ 5 \sqrt{5} } ( 5 i + 6 j - 8k ). #

# \ #

# "Summarizing:" #

# "Solution vector": " \quad \quad \quad \quad \quad \hat{D} \ = \ 1/{ 5 \sqrt{5} } ( 5 i + 6 j - 8k ). #

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