What are the coordinates of a point in three-dimensional space?

The coordinates of a point in three-dimensional space are represented as (x, y, z), where x is the distance along the x-axis, y is the distance along the y-axis, and z is the distance along the z-axis.

How do you find the distance between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in 3D space?

The distance d can be calculated using the formula: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²).

What is the formula for the midpoint of the segment joining the points A(x₁, y₁, z₁) and B(x₂, y₂, z₂)?

The midpoint M is given by M = ((x₁ + x₂)/₂, (y₁ + y₂)/₂, (z₁ + z₂)/₂).

How do you represent a line in 3D that passes through point A and is parallel to vector B?

The equation of the line can be represented as r = A + λB, where A is the position vector of point A, B is the direction vector, and λ is a scalar parameter.

What is the section formula in three-dimensional geometry?

What are direction cosines and how are they calculated?

Direction cosines are the cosines of the angles that a line makes with the coordinate axes. If α, β, and γ are the angles, then the direction cosines are l = cos(α), m = cos(β), n = cos(γ). They satisfy the relation l² + m² + n² = 1.

If a line has direction cosines l = 1/√3, m = 1/√3, and n = 1/√3, what does it indicate about the angles with the axes?

It indicates that the line makes equal angles with the x-axis, y-axis, and z-axis, specifically angles of 45 degrees, since cos(45°) = 1/√2.

How do you find the volume of a rectangular prism with length l, width w, and height h?

The volume V of the rectangular prism is calculated using the formula V = l × w × h.

What is the formula for the surface area of a cylinder?

The surface area A of a cylinder is given by A = 2πr(h + r), where r is the radius and h is the height of the cylinder.

How can you determine if three points A, B, and C are collinear in 3D space?

Three points A, B, and C are collinear if the vectors AB and AC are parallel, which can be checked by seeing if the cross product of AB and AC is the zero vector.

If a point P(1, 2, 3) is reflected across the plane x + y + z = 6, what are the coordinates of the reflected point?

The reflected point P' can be found using the formula for reflection across a plane. The coordinates are P' = (5, 4, 3).

What is the equation of a plane defined by the normal vector (a, b, c) and passing through the point (x₀, y₀, z₀)?

The equation of the plane is given by a(x - x₀) + b(y - y₀) + c(z - z₀) = 0.

What are the coordinates of the centroid of a triangle with vertices at A(x₁, y₁, z₁), B(x₂, y₂, z₂), and C(x₃, y₃, z₃)?

The coordinates of the centroid G are given by G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3).

Calculate the volume of a sphere with radius r. 

Hint: Use the formula V = (4/3)πr³.

The volume V of a sphere is V = (4/3)πr³. For example, if r = 3, then V = (4/3)π(3)³ = 36π.

If a line passes through points A(1, 1, 1) and B(4, 5, 6), what are the direction ratios of the line?

The direction ratios can be found by subtracting the coordinates: (4 - 1, 5 - 1, 6 - 1) = (3, 4, 5).

What is the formula for the distance between a point and a plane?

The distance d from a point (x₀, y₀, z₀) to the plane Ax + By + Cz + D = 0 is given by d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²).

What is the equation of a line joining points A(0, 0, 0) and B(1, 2, 3)? 

Hint: Use parametric equations.

The parametric equations of the line are x = t, y = 2t, z = 3t, where t is a scalar.

What is the relationship between direction ratios and direction cosines?

Direction ratios are proportional to direction cosines. If l, m, n are direction cosines, then the direction ratios a, b, c can be expressed as a = kl, b = km, c = kn for some constant k.