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Date May 2019 Marks available 7 Reference code 19M.1.SL.TZ1.S_6
Level Standard Level Paper Paper 1 Time zone Time zone 1
Command term Find Question number S_6 Adapted from N/A

Question

The magnitudes of two vectors, u and v, are 4 and  3  respectively. The angle between u and v is  π 6 .

Let w = u − v. Find the magnitude of w.

Markscheme

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

METHOD 1 (cosine rule)

diagram including u, v and included angle of  π 6       (M1)

eg   

sketch of triangle with w (does not need to be to scale)      (A1)

eg  

choosing cosine rule      (M1)

eg     a 2 + b 2 2 a b cos C

correct substitution        A1

eg    4 2 + ( 3 ) 2 2 ( 4 ) ( 3 ) cos π 6

cos π 6 = 3 2  (seen anywhere)       (A1)

correct working        (A1)

eg    16 + 3 − 12

| w | =  7         A1    N2

 

METHOD 2 (scalar product)

valid approach, in terms of u and v (seen anywhere)      (M1)

eg   | w |2 = (u − v)•(u − v), | w |2 = u− 2uvv, | w |= ( u 1 v 1 ) 2 + ( u 2 v 2 ) 2 ,

| w | =  ( u 1 v 1 ) 2 + ( u 2 v 2 ) 2 + ( u 3 v 3 ) 2

correct value for uu (seen anywhere)       (A1)

eg   | u|2 = 16,  uu = 16,  u 1 2 + u 2 2 = 16

correct value for vv (seen anywhere)      (A1)

eg  | v|2 = 16,  vv = 3,  v 1 2 + v 2 2 + v 3 2 = 3

cos ( π 6 ) = 3 2   (seen anywhere)      (A1)

uv  = 4 × 3 × 3 2   (= 6)  (seen anywhere)       A1

correct substitution into u− 2uvv or u 1 2 + u 2 2 + v 1 2 + v 2 2 2 ( u 1 v 1 + u 2 v 2 )   (2 or 3 dimensions)      (A1)

eg   16 − 2(6) + 3  (= 7)

| w | =  7         A1    N2

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.13—Scalar and vector products
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Topic 3—Geometry and trigonometry

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