Planes and the Vector Product flashcards
26 practice cards drawn from the Planes and the Vector Product lessons. Tap a card to turn it over. Every answer is checked against the lesson it came from.
Read the Planes and the Vector Product lessons in full →
(2, 4, 0) × (4, 3, 0)
(0, 0, -10)
from “The Vector Product”
(4, 1, 0) × (1, 1, 0)
(0, 0, 3)
from “The Vector Product”
Expand and simplify (a + 4b) × (a − b).
−5(a × b)
from “Expanding Products of Vectors”
|a| = 2, |b| = 2 and a · b = 1. What is ?
10
from “Expanding Products of Vectors”
a × b has length 16. What is the area of the triangle with sides a and b?
8
from “Parallelogram Area from the Vector Product”
a × b = (3, 4, 12). What is the area of the parallelogram a and b span?
13
from “Parallelogram Area from the Vector Product”
Can (1, 3, 1) and (3, 9, 3) be the two directions of a plane?
no
from “The Vector Equation of a Plane”
Can (1, 3, 3) and (3, 9, 10) be the two directions of a plane?
yes
from “The Vector Equation of a Plane”
Which vector is normal to 3x − 1y + 3z = 6?
(3, -1, 3)
from “The Cartesian Equation of a Plane”
Which vector is normal to 3x − 4y + 4z = 6?
(3, -4, 4)
from “The Cartesian Equation of a Plane”
Which one gives the angle between a line of direction d and a plane of normal n?
from “The Angle Between a Line and a Plane”
A line meets the normal of a plane at 50°. What angle does it make with the plane?
40°
from “The Angle Between a Line and a Plane”
Which pair of vectors gives the angle between two planes?
their two normals
from “The Angle Between Two Planes”
Two planes have normals (3, 1, 3) and (6, 3, 6). Are the planes parallel?
no
from “The Angle Between Two Planes”
Substituting a line into a plane gives 0t = 0. What does that say?
the line lies inside the plane
from “The Intersection of a Line and a Plane”
Substituting the line into the plane gives 5t + 3 = 23. What is t?
4
from “The Intersection of a Line and a Plane”
P is (6, 6, 4) and sits 3 from a plane with normal (2, 1, 2), which has length 3. Where is the foot of the perpendicular?
(4, 5, 2)
from “The Distance from a Point to a Plane”
Which direction gives the shortest route from a point to a plane?
along the normal
from “The Distance from a Point to a Plane”
A line passes through (2, 1, 0) in the direction (2, 1, 2), which has length 3. Where is the foot of the perpendicular from P = (7, 5, 2)?
(6, 3, 4)
from “The Distance from a Point to a Line”
A line passes through (1, 2, 2) in the direction (2, 3, 6), which has length 7. Where is the foot of the perpendicular from P = (11, 10, 11)?
(5, 8, 14)
from “The Distance from a Point to a Line”
Which point lies on ?
(5, 2, 3)
from “The Cartesian Form of a Line”
Write r = (2, 3, 3) + t(5, 3, 3) in Cartesian form.
from “The Cartesian Form of a Line”
Equating two lines in space gives how many equations?
3
from “Intersecting and Skew Lines”
Are these two lines parallel, intersecting or skew?
skew
from “Intersecting and Skew Lines”
Setting z = 0 leaves x + y = 6 and x + 2y = 14. Which point is on the line?
(−2, 8, 0)
from “The Intersection of Two Planes”
Two planes that are not parallel share what?
a line
from “The Intersection of Two Planes”