A-LevelFurther MathematicsVectors9231/12 · Oct/Nov 2023 · Q513 Marks

The plane Π₁ has equation r = i− j − 2k + \lambda(i−2j − 3k) + \mu(3i− k).

(a)Find an equation for Π₁ in the form ax+by+cz = d.[4]
(b)The line l, which does not lie in Π₁, has equation r = -3i+k+t(i+j+k). Show that l is parallel to Π₁.[2]
(c)Find the distance between l and Π₁.[3]
(d)The plane Π₂ has equation 3x+3y+2z = 1. Find a vector equation of the line of intersection of Π₁ and Π₂.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[M1 A1] Finds perpendicular to Π₁. `n = (1, -2, -3) × (3, 0, -1) = (2, -8, 6)`.
[M1 A1] Uses point on Π₁. `2x - 8y + 6z = 2(1) - 8(-1) + 6(-2) = -2`. Simplifies to `x - 4y + 3z = -1`.
[M1 A1] Shows dot product with direction of line is 0. `(1, 1, 1) · (1, -4, 3) = 1 - 4 + 3 = 0`.
[M1 A1 A1] Uses correct formula for distance from point on l to Π₁. `D = (|(1)(-3) + (-4)(0) + (3)(1) - (-1)|)/(√(1²+(-4)²+3²)) = 1/√26`.
[B1] States point common to both planes e.g. `(1/15, 4/15, 0)` or `(5/7, 0, -4/7)` or `(0, 5/17, -7/17)` or alternative.
[M1 A1] Finds direction of line. `n₁ × n₂ = (1, -4, 3) × (3, 3, 2) = (-17, 7, 15)`.
[A1] OE. `r = (1/15 i + 4/15 j) + \lambda(-17i + 7j + 15k)`.

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