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Date May 2019 Marks available 6 Reference code 19M.2.SL.TZ2.S_7
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number S_7 Adapted from N/A

Question

The vector equation of line L is given by r  = ( 1 3 8 ) + t ( 4 5 1 ) .

Point P is the point on L that is closest to the origin. Find the coordinates of P.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1 (Distance between the origin and P)

correct position vector for OP       (A1)

eg  OP = ( 1 + 4 t 3 + 5 t 8 t ) P = ( 1 + 4 t , 3 + 5 t , 8 t )

correct expression for OP or OP2 (seen anywhere)       A1

eg    ( 1 + 4 t ) 2 + ( 3 + 5 t ) 2 + ( 8 t ) 2 ( 1 + 4 x ) 2 + ( 3 + 5 x ) 2 + ( 8 x ) 2

valid attempt to find the minimum of OP       (M1)

eg    d = 0 , root on sketch of d ,  min indicated on sketch of  d

t = 1 14 , 0.0714285       (A1)

substitute their value of t into L (only award if there is working to find t )       (M1)

eg   one correct coordinate,   1 + 4 ( 1 14 )

( 1.28571 , 2.64285 , 8.07142 )

( 9 7 , 37 14 , 113 14 ) = ( 1.29 , 2.64 , 8.07 )        A1  N2

 

METHOD 2 (Perpendicular vectors)

recognizing that closest implies perpendicular     (M1)

eg  OP L   (may be seen on sketch),  a b = 0

valid approach involving  OP        (M1)

eg    OP = ( 1 + 4 t 3 + 5 t 8 t ) , ( 4 5 1 ) OP , ( 4 5 1 ) OP

correct scalar product        A1

eg    4 ( 1 + 4 t ) + 5 ( 3 + 5 t ) 1 ( 8 t ) ,   4 + 16 t + 15 + 25 t 8 + t = 0 ,   42 t + 3

t = 1 14 , 0.0714285       (A1)

substitute their value of t into L or OP (only award if scalar product used to find t )      (M1)

eg   one correct coordinate,   1 + 4 ( 1 14 )

( 1.28571 , 2.64285 , 8.07142 )

( 9 7 , 37 14 , 113 14 ) = ( 1.29 , 2.64 , 8.07 )        A1  N2

 

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.11—Vector equation of a line in 2d and 3d
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Topic 3—Geometry and trigonometry » AHL 3.13—Scalar and vector products
Topic 3—Geometry and trigonometry

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