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Vector Algebra

Question
CBSEENMA12035716

Find the distance of the point (2, 12, 5) from the point of intersection of the line 
straight r with rightwards arrow on top equals 2 straight i with hat on top minus 4 straight j with hat on top plus 2 straight k with hat on top plus straight lambda open parentheses 3 straight i with hat on top plus 4 straight j with hat on top plus 2 straight k with hat on top close parentheses space and space the space plane space straight r with rightwards arrow on top. open parentheses straight i with hat on top minus 2 straight j with hat on top plus straight k with hat on top close parentheses equals 0.

Solution

Any point in the line is
2 plus 3 straight lambda comma space minus 4 plus 4 straight lambda comma space space 2 plus 2 straight lambda
The vector equation of the plane is given as
straight r with rightwards arrow on top. space open parentheses straight i with hat on top minus 2 straight j close parentheses plus straight k right parenthesis space equals space 0
Thus space the space cartesian space equation space of space the space plane space is space straight x minus 2 straight y plus straight z equals space 0
Since space the space point space lies space in space the space plane
left parenthesis 2 plus 3 straight lambda right parenthesis 1 plus left parenthesis negative 4 plus 4 straight lambda right parenthesis thin space left parenthesis negative 2 right parenthesis plus left parenthesis 2 plus 2 straight lambda right parenthesis 1 space equals 0
rightwards double arrow space 2 plus 8 plus 2 plus 3 straight lambda minus 8 straight lambda plus 2 straight lambda equals 0
rightwards double arrow 12 minus 3 straight lambda equals 0
rightwards double arrow 12 equals space 3 straight lambda
rightwards double arrow straight lambda equals 4
Thus, the point of intersection of the line and the plane is:
2 plus 3 cross times 4 comma space minus 4 plus 4 cross times 4 comma space space 2 plus 2 cross times 4
rightwards double arrow 14 comma space 12 comma space 10
Distance between (2, 12, 5) and (14, 12, 10) is:
straight d equals square root of left parenthesis 14 minus 2 right parenthesis squared plus left parenthesis 12 minus 12 right parenthesis squared plus left parenthesis 10 minus 5 right parenthesis squared end root
rightwards double arrow straight d equals space square root of 144 plus 25 end root
rightwards double arrow space straight d equals square root of 169
rightwards double arrow straight d space equals space 13 space units