In 3D space, a point is written using how many coordinates
A One coordinate
B Two coordinates
C Four coordinates
D Three coordinates
In three-dimensional geometry, a point is represented as an ordered triple (x, y, z). Each number shows the directed distance from the origin along the x-, y-, and z-axes.
A point lies on the z-axis when which condition holds
A x = 0, y = 0
B y = 0, z = 0
C x = 0, z = 0
D x = y, z = 0
Points on the z-axis have no displacement along x or y. So x and y must both be zero, while z can be any real number, positive or negative.
Point (0, 5, 0) lies on which axis
A x-axis
B y-axis
C z-axis
D No axis
A point lies on the y-axis when x=0 and z=0. For (0,5,0), both conditions hold, so it is on the y-axis, at distance 5 from origin.
Coordinate planes divide space into octants because they intersect at
A A line only
B A circle only
C A random point
D The origin
The three coordinate planes x=0, y=0, z=0 intersect at the origin (0,0,0). Their intersection divides space into 8 regions based on the signs of x, y, and z.
If direction ratios are (2, −2, 1), one possible proportional set is
A (2, 2, 1)
B (−2, −2, 1)
C (4, −4, 2)
D (1, 1, 1)
Direction ratios can be multiplied by any nonzero constant without changing direction. Multiplying (2, −2, 1) by 2 gives (4, −4, 2), which represents the same line direction.
If direction cosines are (l, m, n), then l represents
A cos with y-axis
B cos with z-axis
C sin with x-axis
D cos with x-axis
Direction cosines are defined as cosines of angles a line makes with the positive coordinate axes. So l = cosα where α is the angle with x-axis.
When a line makes 90° with x-axis, its direction cosine l equals
A 0
B 1
C −1
D 1/2
If the angle with x-axis is 90°, then l = cos90° = 0. This means the line has no component along the x-direction in its unit direction vector.
A line through (1,2,3) and (4,6,3) has direction ratios
A (−3,−4,0)
B (3,4,0)
C (4,6,3)
D (5,8,6)
Direction ratios from two points are found using coordinate differences: (x2−x1, y2−y1, z2−z1). Here they are (4−1, 6−2, 3−3) = (3,4,0).
For points P(1,0,0) and Q(1,0,5), distance PQ equals
A √5 units
B √26 units
C 6 units
D 5 units
Only z-coordinate changes from 0 to 5; x and y stay same. So distance is √(0²+0²+5²)=5. The segment is parallel to z-axis.
The distance squared between (x1,y1,z1) and (x2,y2,z2) is
A Δx²+Δy²+Δz²
B (Δx+Δy+Δz)²
C Δx²Δy²Δz²
D √(Δx²+Δy²+Δz²)
Before taking square root, the squared distance equals (x2−x1)²+(y2−y1)²+(z2−z1)². This form is useful to avoid radicals in comparisons.
If a point divides PQ in ratio 1:2 internally, it is
A Closer to P
B Midpoint always
C Closer to Q
D Outside segment
In internal division m:n, the point is closer to the endpoint with smaller weight. For 1:2, weight near P is 1, so the point lies closer to Q.
Equation z = 0 represents
A yz-plane
B zx-plane
C z-axis
D xy-plane
The xy-plane consists of all points where z-coordinate is zero. x and y can vary freely, so z=0 is the plane containing x- and y-axes.
Equation x = 0 represents
A xy-plane
B yz-plane
C zx-plane
D x-axis
The yz-plane contains y- and z-axes and has x-coordinate zero for every point. So x=0 is the standard equation of the yz-plane.
The vector form of a line is written as
A r = a + tb
B r = a · b
C r = a × b
D r = a − b
In vector form, r is the position vector of any point on the line. a is a fixed point vector and b is direction vector. Parameter t moves along the line.
If two lines have direction vectors u and v, then cosθ equals
A |u×v|/|u||v|
B |u|/|v|
C (u·v)/|u×v|
D (u·v)/|u||v|
The angle between two lines is computed from their direction vectors using dot product. Dot product relates directly to cosθ, giving the standard formula for angle.
If a plane is parallel to yz-plane, its equation looks like
A y = constant
B z = constant
C x = constant
D x+y+z = 0
yz-plane is x=0. Any plane parallel to it must keep same normal direction (along x-axis), so it will be of form x = k where k is constant.
Plane equation ax+by+cz+d=0 is called
A Linear plane form
B Quadratic surface
C Sphere form
D Cylindrical form
A plane in 3D is represented by a first-degree (linear) equation in x, y, z. Higher-degree equations represent curved surfaces like spheres or cones.
If a plane passes through origin, then in ax+by+cz+d=0
A a = 0
B b = 0
C c = 0
D d = 0
Substituting origin (0,0,0) into the plane equation gives d=0. So planes through origin have form ax+by+cz=0 with nonzero normal vector.
The intercept form x/a + y/b + z/c = 1 requires
A a=b=c only
B a,b,c nonzero
C d must be zero
D plane through origin
In intercept form, a, b, c are intercepts on axes. If any intercept is zero, the plane passes through origin on that axis and the form breaks due to division by zero.
For a plane, the normal vector gives
A Perpendicular direction
B Parallel direction
C Midpoint location
D Axis intercepts
The normal vector of a plane is perpendicular to the plane surface. It helps to find angles between planes and test whether a line is parallel or perpendicular to the plane.
Two planes are parallel when their normals are
A Perpendicular vectors
B Zero vectors
C Parallel vectors
D Unit vectors only
Planes are parallel if they have the same orientation. This happens when their normal vectors are proportional, meaning (a,b,c) is a scalar multiple of (a′,b′,c′).
Two planes are perpendicular when their normals satisfy
A Cross product zero
B Same magnitude
C Same direction
D Dot product zero
Angle between planes equals angle between their normals. If planes are perpendicular, normals are perpendicular, so their dot product becomes zero.
A line is parallel to plane ax+by+cz+d=0 if its direction vector v satisfies
A v · (a,b,c)=0
B v × (a,b,c)=0
C v = (a,b,c)
D |v| = 0
A line is parallel to a plane when its direction is perpendicular to the plane’s normal. So dot product of line direction vector with plane normal must be zero.
A line is perpendicular to plane ax+by+cz+d=0 if its direction is
A Perpendicular to normal
B Parallel to normal
C Parallel to plane
D In plane only
A line perpendicular to a plane must go along the plane’s normal direction. If the line direction vector is proportional to (a,b,c), it makes 90° with the plane.
Equation of line through two points uses which direction vector
A P + Q
B P − Q + 1
C P × Q
D Q − P
The line through P and Q is directed from P to Q. So the direction vector is Q−P, obtained by subtracting coordinates of P from coordinates of Q.
If direction ratios are (a,b,c), then a valid direction vector is
A (a,b,c)
B (a+b+c)
C (ab,bc,ca)
D (a²,b²,c²)
Direction ratios themselves represent components of a direction vector. Any nonzero scalar multiple also works, but (a,b,c) is the simplest direction vector form.
In 3D, dot product of perpendicular vectors equals
A 1
B −1
C 0
D product of magnitudes
When two vectors are perpendicular, the angle between them is 90°. Since u·v = |u||v|cosθ, and cos90°=0, the dot product becomes zero.
A line with direction ratios (0,0,5) is parallel to
A x-axis
B y-axis
C xy-plane
D z-axis
Direction ratios (0,0,5) indicate movement only along z direction. Any nonzero multiple of (0,0,1) represents the z-axis direction, so the line is parallel to z-axis.
If direction cosines are (−l, −m, −n), the direction is
A Opposite direction
B Same direction
C Perpendicular always
D Undefined always
Negating direction cosines reverses the direction vector. The line remains the same geometrically, but the direction is opposite, meaning it points into the opposite octant.
The angle between a line and x-axis is 0° when the line is
A Parallel to y-axis
B Parallel to x-axis
C Parallel to z-axis
D In yz-plane
Angle 0° with x-axis means the line points exactly along positive x direction. Then its direction cosine l=cos0°=1 and other components become zero.
A point (a,b,c) reflected in origin becomes
A (a,−b,c)
B (a,b,−c)
C (−a,b,c)
D (−a,−b,−c)
Reflection in origin changes the sign of every coordinate because the origin is the midpoint between the point and its image. So (a,b,c) maps to (−a,−b,−c).
A plane parallel to xy-plane at height 7 has equation
A z = 7
B y = 7
C x = 7
D x+y = 7
Planes parallel to xy-plane have constant z value. “Height 7” means z-coordinate is 7 everywhere on the plane, so equation is z = 7.
The distance from point (2,0,0) to y-axis equals
A 0
B √2
C 2
D 4
Points on y-axis have x=0 and z=0. Closest point to (2,0,0) on y-axis is (0,0,0). Distance becomes √((2−0)²+0²+0²)=2.
A line in symmetric form (x−1)/2 = (y−2)/3 = (z−0)/1 passes through
A (2,3,1)
B (0,0,0)
C (1,0,2)
D (1,2,0)
In symmetric form, the point (x1,y1,z1) is the point where numerators become zero. Here x=1, y=2, z=0 satisfies the line, so it passes through (1,2,0).
For line x=1+2t, y=2+3t, z=t, direction ratios are
A (1,2,3)
B (2,3,1)
C (1,3,2)
D (2,1,3)
In parametric form x=x1+at, y=y1+bt, z=z1+ct, the coefficients of t give the direction ratios. Here they are 2, 3, and 1.
If two lines have direction vectors u and v, they are parallel when
A u × v = 0
B u·v = 0
C |u| = |v|
D u+v = 0 always
Cross product of two vectors is zero when they are parallel or anti-parallel. So if direction vectors have zero cross product, the lines are parallel (or same line direction).
Shortest distance between two parallel lines in 3D is measured along
A Any joining line
B Direction vector
C Coordinate plane
D Perpendicular segment
Parallel lines have a constant separation. The shortest distance is the length of a segment perpendicular to both lines, connecting one line to the other.
If point P lies on plane ax+by+cz+d=0, then
A ax+by+cz+d = 1
B ax+by+cz+d = 0
C ax+by+cz+d > 0
D ax+by+cz+d < 0
A point lies on a plane if its coordinates satisfy the plane equation exactly. Substituting (x,y,z) into ax+by+cz+d must give zero.
If a plane contains x-axis, then which points lie on it
A (t,0,0)
B (0,t,0)
C (0,0,t)
D (t,t,t)
x-axis consists of all points where y=0 and z=0, with x variable. So any plane containing x-axis must contain every point (t,0,0).
A plane through three non-collinear points is
A Many planes
B No plane exists
C Always parallel axes
D Unique plane
Three non-collinear points determine exactly one plane in 3D. If points were collinear, infinitely many planes could pass through that line, but non-collinear fixes it uniquely.
Two lines in 3D are skew when they are
A Parallel and coplanar
B Intersecting lines
C Non-parallel, non-coplanar
D Perpendicular and coplanar
Skew lines do not intersect and are not parallel, and they lie in different planes. That is why no single plane can contain both skew lines.
If u·v is negative, then the angle between vectors is
A Obtuse angle
B Acute angle
C Right angle
D Zero angle
Since u·v = |u||v|cosθ, a negative dot product means cosθ is negative. That occurs when θ is between 90° and 180°, which is an obtuse angle.
The distance from point (0,3,4) to origin equals
A 7
B √7
C 12
D 5
Distance from origin is √(x²+y²+z²). Here √(0²+3²+4²)=√(9+16)=√25=5, a classic 3–4–5 right triangle in 3D.
A line intersects a plane at one point when line is
A Parallel to plane
B Not parallel plane
C Lying in plane
D Perpendicular axes
If a line is not parallel to a plane, it meets the plane at exactly one point (unless it lies entirely in the plane). Parallel lines have no intersection point with the plane.
If a line lies in a plane, then every point of line
A Fails plane equation
B Makes sphere
C Satisfies plane equation
D Makes cone
A line contained in a plane means all its points are also on the plane. So substituting any point from the line into the plane equation must yield zero.
The image of point P in a plane is obtained by
A Rotation about axis
B Translation along z
C Scaling from origin
D Reflection across plane
“Image” of a point in a plane usually means reflection: the plane becomes the perpendicular bisector of segment joining the point and its image. Distances to plane are equal and opposite.
Equation (x−1)²+(y−2)²+(z−3)²=9 represents
A Sphere radius 3
B Plane equation
C Cylinder equation
D Line equation
Standard sphere form is (x−a)²+(y−b)²+(z−c)²=r². Here center is (1,2,3) and r²=9, so radius is 3.
A cylinder in 3D is formed when one coordinate is
A Fixed constant
B Always negative
C Always zero
D Not involved equation
If an equation involves only two variables, the third is free, creating a surface extended along that axis. For example, x²+y²=r² is a cylinder along z-axis.
A cone in 3D generally has equation involving
A First degree only
B Second degree terms
C Only constants
D Only sine terms
A cone is a quadratic surface, typically involving squared terms like x², y², z². For example, x²+y²=z² is a right circular cone with vertex at origin.
Coordinate transformation by translation changes
A Origin shifts only
B Distances change
C Angles change
D Shape changes
Translation moves the origin to a new point, changing coordinate values of points. But distances, angles, and shapes remain unchanged because translation is a rigid motion in space.