Zadanie 11b
Find the centre of C.
Plane intersections and angles
Determine intersections involving lines and planes, interpret the resulting solution sets and calculate relevant angles.
Tę umiejętność sprawdza 28 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 28 Otwórz w wyszukiwarce z filtrami
Zadanie 11e
Let . Find its acute angle with Π to the nearest degree.
Zadanie 11a
(i) Show that a=1. (ii) Determine the possible real values of b.
Zadanie 11b
Now b=2. Line L₂ passes through A with direction (2,1,−1). Find its intersection N with Π.
Zadanie 11c
The acute angle of L₂ with Π is α. Find sinα exactly.
Zadanie 11d
Given AN=3√6, find the perpendicular distance from L₁ to Π.
Zadanie 11e
Find a vector equation for the reflection of L₁ in Π.
Zadanie 9b
Three planes are P1 : 2x − y + z = 0, P2 : x + by + 2z = 5 and P3 : 3x + y + 3z = d, where b, d > 0. They have a common line. Find b and d.
Zadanie 11a
Show that the acute angle between the two panels is 60°.
Zadanie 11e
Plane P2 is 2x + y + z = 4. Prove that the acute angle θ between P1 and P2 satisfies .
Zadanie 11f
A line through E perpendicular to P1 meets P2 : 2x + y + z = 4 at F. Find F.
Zadanie 11b
Suppose p=−2, Π₁ and Π₂ are perpendicular, and A lies on Π₂. Find k and q.
Zadanie 11c
Suppose k=2 and Π₂ is parallel to Π₁. Find p.
Zadanie 11d
Suppose k=2, p=−3/2 and q=75/2. The normal line to Π₁ through A meets Π₂ at B. (i) Find B. (ii) Show that the distance between the planes is .
Zadanie 11e
The plane 2x + 4y − z = 25 is at distance from Π₁. Find the other plane Π₃ parallel to Π₁ at the same distance.
Zadanie 11d
A line M is parallel to Π, passes through (2,1,3), and intersects the z-axis. Find a vector equation of M.
Zadanie 6a
Verify that , , describes L.
Zadanie 8
The acute angle between line (x − 2)/2 = y + 1 = (3 − z)/2 and plane is α, with 0<α<π/2. Find α in radians.
Zadanie 12b
Verify that both lines lie in P.
Zadanie 12c
(i) Find C. (ii) Hence find BC.
Zadanie 12d
Find the reflection of B in P. You may use C=(9/2,5/2,3) as the perpendicular foot.
Zadanie 12b
Verify that both lines lie in P.
Zadanie 12c
(i) Find C. (ii) Hence find BC.
Zadanie 12d
Find the reflection of B in P. You may use C=(9/2,5/2,3) as the perpendicular foot.
Zadanie 11a
Show that no point lies on all three planes.
Zadanie 11b
(i) Verify that P lies on both Π₁ and Π₂. (ii) Find a vector equation of their line of intersection L.
Zadanie 11c
Let L be the line of intersection of Π₁ and Π₂. Find the distance between L and Π₃.
Zadanie 12d
For k≠8, let Π be the plane through A,B,C. (i) Find its Cartesian equation. (ii) Find the point of Π nearest the origin.