Vectors VI Lowersixth Science MathematicsOnline version
Quick true/false quiz on planes and angles.
1
The vector form of a plane through P with normal n is n·(r-P)=0, where r=(x,y,z).
2
The vector form of the plane equation uses the position vector r and the normal vector n.
3
If a plane passes through the origin, its normal vector must be zero.
4
Intercept form x/a + y/b + z/c = 1 is valid only if a,b,c are equal.
5
A plane parallel to ax+by+cz=d has a different left-hand side from ax+by+cz=d'.
6
The cross product of two direction vectors lying in a plane is a normal vector to that plane.
7
The intercept form x/a + y/b + z/c = 1 can be used even if any of a, b, or c is zero.
8
The angle between a line and a plane is the complement of the angle between the line and the plane's normal.
9
If the line L is contained in a plane, the direction vector of L is orthogonal to the plane's normal.
10
For the plane 3x - y + 2z = 12, the intercept on the x-axis is 4.
11
To determine d in ax+by+cz=d for a plane through a point P, substitute P into the equation to solve for d.
12
A plane containing the point (1,2,3) with normal vector (1,0,-1) has equation (x-1) - (z-3) = 0.
13
A plane with equation ax+by+cz=d is parallel to the plane ax+by+cz=d' if d ≠ d'.
14
If a line is perpendicular to a plane, the angle between the line and the plane is 0 degrees.
15
For a plane ax+by+cz=d, the intercepts on the axes occur where two variables are zero and the remaining variable equals d divided by the corresponding coefficient.
16
If a line is parallel to a plane, the angle between the line and the plane is 0 degrees.
17
The angle between a line and a plane is always equal to the angle between the line and the plane normal.
18
Substituting a known point on the plane into ax+by+cz=d verifies the plane equation.
19
A plane is parallel to another plane if their normal vectors are proportional.
20
The equation x/2 + y/3 + z/6 = 1 has intercepts 2, 3, and 6 on the axes.
21
The condition for parallel planes is that their normals are proportional and the constants differ.
22
The dot product n·(r-P) equals zero for all points r lying in the plane through P with normal n.
23
If two planes have different d values in ax+by+cz=d, they are necessarily not the same plane.
24
The equation of a plane cannot be written using a point and a normal vector.
25
If a plane contains a point P and is parallel to another plane with equation ax+by+cz=d2, then the two planes share the same normal (a,b,c).
26
A plane is uniquely determined by a point and a non-parallel normal vector to the plane.
27
If a line is perpendicular to a plane, the angle between the line and the plane is 90 degrees.
28
If a plane has normal vector n and passes through P, its equation is n·(r-P)=0.
29
If v is parallel to the plane, then v is orthogonal to the plane’s normal vector.
30
The angle between a line and a plane equals zero if the line lies inside the plane.
31
The equation ax+by+cz=d is the standard form of a plane in 3D.
32
A plane containing P and parallel to a given plane must share the same normal vector as that plane.
33
The normal form of a plane equation is derived from a point and the plane's normal vector.
34
The equation of a plane containing point P(x0,y0,z0) and normal n=(a,b,c) can be written as a(x-x0)+b(y-y0)+c(z-z0)=0.
35
If a plane contains the points P and Q, the vector PQ lies in the plane and is orthogonal to the plane's normal.
36
Two parallel planes must have completely different normal vectors.
37
If a plane's normal is (1,2,3) and it passes through the point (0,0,0), then the plane equation is x+2y+3z=0.
38
The dot product n·(r-P) is never used in plane equations.
39
The normal vector of the plane ax+by+cz=d is n=(a,b,c).
40
A plane through P with normal n has symmetric form (r-P)·n=0.
41
A plane parallel to the plane 2x - y + 3z = 4 has an equation of the form 2x - y + 3z = k for some k.
42
The distance from the origin to the plane ax+by+cz=d is |d|/sqrt(a^2+b^2+c^2) when the plane passes through the origin.
43
The equation of a plane can be determined if we know a point on the plane and a normal vector.
44
A plane containing a given point P and parallel to a given plane has the same normal vector as the given plane.
45
The angle between a line with direction vector v and a plane with normal n satisfies sin(theta) = |n·v|/(||n|| ||v||).
46
A plane parallel to a given plane with equation ax+by+cz=d has the same left-hand side ax+by+cz when written in standard form.
47
The intercept form of a plane is x/a + y/b + z/c = 1 provided a,b,c are the x-, y-, z-intercepts respectively and nonzero.
48
If a plane has normal vector n and passes through P, then P lies on the plane.
49
The angle between a line and a plane is obtained by the arctangent of |n·v| divided by the product of norms.
50
If the normal vector of a plane is perpendicular to a given vector, the angle between the plane and the vector is 0 degrees.
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