O-LevelAdditional MathematicsVectors in two dimensions4037/22 · May/June 2019 · Q108 Marks

Find the unit vector in the direction of `5i – 15j`.

(a)Find the unit vector in the direction of `5i – 15j`.[2]
(b)(i)The position vectors of points A and B relative to an origin O are `(3; -5)` and `(12; 7)` respectively. The point C lies on AB such that AC : CB is 2 : 1. Find the position vector of C relative to O.[3]
(b)(ii)The point D lies on OB such that OD : OB is `1 : λ` and `DC = (6; 1.25)`. Find the value of `λ`.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[M1] (a) √(5²+(-15)²) seen
[A1] (1/√250) * (5i - 15j) oe, isw
[B1] (b)(i) (12; 7) oe, soi
[M1] (1/3)*(3; -5) + (2/3)*(12; 7) oe, soi OR (3; -5) + (2/3) * ( (12; 7) - (3; -5) )
[A1] (9; 3)
[M1] (b)(ii) Forms a valid vector relationship using DC and e.g. OD or DB. e.g. (6; 1.25) = OC - OD = (9; 3) - (1/λ)(12; 7) oe
[A1] Finds a correct proportion e.g. OB = λOD oe soi OR 3OB = λDB oe soi OR solves a correct equation in λ. e.g. 6 = 9 - (1/λ)*12
[A1] λ = 4 from a fully correct method

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