Three-dimensional geometry: lines and planes — IB Maths AA practice questions

Question 1

Are the four points $(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)$ coplanar?

Accepted Answer

The tetrahedron with these vertices has volume $1/6 
e 0$, so the four points are NOT coplanar — they form a non-degenerate tetrahedron.

Answer

Yes

Answer

No

Answer

Only three of them are

Answer

Cannot determine

Question 2

The volume of the tetrahedron with vertices $(0,0,0), (3,0,0), (0,4,0), (0,0,5)$ is:

Accepted Answer

The three edges from origin are $(3,0,0), (0,4,0), (0,0,5)$. Determinant: $3 \cdot 4 \cdot 5 = 60$. Volume: $60/6 = 10$.

Answer

$5$

Answer

$10$

Answer

$30$

Answer

$60$

Question 3

For which value of $a$ is $(a, 1, 2) \perp (3, -1, -1)$?

Accepted Answer

$3a - 1 - 2 = 0 \Rightarrow 3a = 3 \Rightarrow a = 1$.

Answer

$0$

Answer

$1$

Answer

$-1$

Answer

$3$

Question 4

The radius of the sphere $(x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 49$ is:

Accepted Answer

$r^2 = 49 \Rightarrow r = 7$.

Answer

$6$

Answer

$7$

Answer

$14$

Answer

$49$

Question 5

The distance from $(0, 0, 0)$ to the plane $2x + 2y + z - 6 = 0$ is:

Accepted Answer

$d = |{-6}|/\sqrt{4 + 4 + 1} = 6/3 = 2$.

Answer

$2$

Answer

$3$

Answer

$\sqrt{5}$

Answer

$6$

Question 6

$(1, 0, 0) 	imes (0, 2, 0) = ?$

Accepted Answer

$(1, 0, 0) 	imes (0, 2, 0) = (0\cdot 0 - 0\cdot 2, 0\cdot 0 - 1\cdot 0, 1\cdot 2 - 0\cdot 0) = (0, 0, 2)$.

Answer

$(0, 0, 0)$

Answer

$(2, 0, 0)$

Answer

$(0, 0, 2)$

Answer

$(0, 2, 0)$

Question 7

The distance from $A(1, 2, 3)$ to $B(4, 6, 3)$ is:

Accepted Answer

$d = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{25} = 5$. (3-4-5 triple in the $xy$-plane.)

Answer

$4$

Answer

$5$

Answer

$7$

Answer

$\sqrt{50}$

Question 8

Midpoint of $A(2, -1, 4)$ and $B(8, 5, -2)$:

Accepted Answer

Average each coordinate: $((2+8)/2, (-1+5)/2, (4-2)/2) = (5, 2, 1)$.

Answer

$(5, 2, 1)$

Answer

$(6, 6, -6)$

Answer

$(3, 3, 1)$

Answer

$(10, 4, 2)$

Question 9

The magnitude $\|\vec{v}\|$ for $\vec{v} = (1, 2, 2)$ is:

Accepted Answer

$\|\vec{v}\| = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$.

Answer

$\sqrt{5}$

Answer

$3$

Answer

$5$

Answer

$9$

Question 10

$(1, 2, 3) \cdot (2, -1, 1) = ?$

Accepted Answer

$1\cdot 2 + 2\cdot(-1) + 3\cdot 1 = 2 - 2 + 3 = 3$.

Answer

$0$

Answer

$2$

Answer

$3$

Answer

$6$

Three-dimensional geometry: lines and planes practice questions — IB Maths AA

Problems & worked solutions

Problem 1 — Three-dimensional geometry: lines and planes

Problem 2 — Three-dimensional geometry: lines and planes

Problem 3 — Three-dimensional geometry: lines and planes

Problem 4 — Three-dimensional geometry: lines and planes

Problem 5 — Three-dimensional geometry: lines and planes

Problem 6 — Three-dimensional geometry: lines and planes

Problem 7 — Three-dimensional geometry: lines and planes

Problem 8 — Three-dimensional geometry: lines and planes

Problem 9 — Three-dimensional geometry: lines and planes

Problem 10 — Three-dimensional geometry: lines and planes