Vector Problems involving Lines & Planes

Exercises for students:
Solve the following problems and check your working using the above applet:

MC N94/II/15 (part)
The equation of the plane p₁ is y + z = 0 and the equation of the line l is r = (5i + 2j + 2k ) + t(2i - j + 3k), where t is a parameter.
Find the position vector of the point of intersection of l and p₁.

MC J88/II/15 (part)
The line l has equation r = (i + 4j - k ) + t(i - j + k) and the plane p has equation r.(-i + 2j + 2k) = 2.
Find the point of intersection of l and p.
Find the acute angle between l and p, correct to the nearest 0.1°.

MC J94/II/15 (part)
The equations of the 2 planes P₁ and P₂ are r.(i + j + 2k) = 2 and r.(i - 3j + 3k) = 3 respectively.
(a) Find the acute angle between the 2 planes, giving your answer to the nearest 0.1°.
(b) Find the length of the projection of the vector i + 4j onto P₁.

MC N96/II/15 (part)
The planes P₁ and P₂ have equations x + y - z = 0 and 2x - 4y + z + 12 = 0 respectively.
Find the coordinates of any one point common to P₁ and P₂, and hence find the equation of the line of intersection of P₁ and P₂, giving your answer in the form r = a + tb.

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