A-LevelFurther MathematicsVectors9231/12 · May/June 2021 · Q612 Marks
Let t be a positive constant. The line l₁ passes through the point with position vector ti+j and is parallel to the vector -2i-j. The line l₂ passes through the point with position vector j+tk and is parallel to the vector -2j+k. It is given that the shortest distance between the lines l₁ and l₂ is √21.
- (a)Find the value of t.[5]
- (b)The plane Π₁ contains l₁ and is parallel to l₂. Write down an equation of Π₁, giving your answer in the form r = a+λb+μc.[1]
- (c)The plane Π₂ has Cartesian equation 5x-6y+7z = 0. Find the acute angle between l₂ and Π₂.[3]
- (d)Find the acute angle between Π₁ and Π₂.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- [B1] Finds vector from any point on l₁ to any point on l₂.
- [M1 A1] Finds common perpendicular.
- [M1 A1] Uses formula for shortest distance.
- [B1 FT] Using their value of t.
- [M1 A1] Uses dot product of -2j+k and normal.
- [A1]
- [M1 A1 FT] Dot product using their normal to Π₁.
- [A1]
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