Vector Addition and Scalar Multiplication

<p>Every point on the line satisfies the plane's equation. (D)</p> Signup and view all the answers

<p>By substituting the parametric equations of the line into the plane's equation. (B)</p> Signup and view all the answers

<p>The line intersects the plane at a unique point. (B)</p> Signup and view all the answers

<p>To produce a vector perpendicular to both of the original vectors. (C)</p> Signup and view all the answers

<p>Choose a point on each line and construct a new line that is not parallel to either. (B)</p> Signup and view all the answers

<p>x = x0 + λa, y = y0 + λb, z = z0 + λc (C)</p> Signup and view all the answers

<p>They are scalar multiples of each other. (C)</p> Signup and view all the answers

<p>$\frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}}$ (A)</p> Signup and view all the answers

<p>ax + by + cz + d = 0 (A)</p> Signup and view all the answers

<p>They are scalar multiples of each other. (A)</p> Signup and view all the answers

<p>By using the dot product formula. (D)</p> Signup and view all the answers

<p>It is a parameter that scales the direction vector. (C)</p> Signup and view all the answers

<p>Three conditions must be satisfied. (A)</p> Signup and view all the answers

<p>A point and a direction vector. (B)</p> Signup and view all the answers

<p>A scalar multiple. (A)</p> Signup and view all the answers

<p>$x - x_0 = rac{y - y_0}{a_2} = rac{z - z_0}{a_3}$ (B)</p> Signup and view all the answers

<p>$PQ = eta a$ for some $eta ho R$ (D)</p> Signup and view all the answers

<p>The corresponding fraction in the Cartesian form is replaced by a constant function. (C)</p> Signup and view all the answers

<p>The angle between vectors determines their parallelism. (A)</p> Signup and view all the answers

<p>It dictates the angle and direction the line extends. (D)</p> Signup and view all the answers

<p>$y - y_0 = a_2 y$ (C)</p> Signup and view all the answers

A line in three-dimensional space is determined by a point and a direction vector.

The set of points (x, y, z) on a line through point P(x0, y0, z0) and vector a(a1, a2, a3) is given by Q(λ) = (x0 + λa1, y0 + λa2, z0 + λa3) with λ ∈ R.

The equation of the line in the form (x - x0)/a1 = (y - y0)/a2 = (z - z0)/a3 = λ

Coordinates (x, y, z) satisfy a line equation if and only if there exists a value of λ such that PQ = λa.

λ represents a real number from negative infinity to infinity that determines a point (x, y, z) on the line.

If any component of 'a' (a1, a2, or a3) is zero, replace the corresponding fraction in the Cartesian equation with the constant function.

The direction vector for the line, it determines the line's direction in 3D space.

Vector connecting point P to point Q (difference of the vectors).

The distance between two points (x1, y1, z1) and (x0, y0, z0) in 3D space is calculated using the formula: √((x1 - x0)² + (y1 - y0)² + (z1 - z0)²).

A point (x, y, z) on a line in 3D space can be expressed as (x₀ + λa, y₀ + λb, z₀ + λc), where (x₀, y₀, z₀) is a point on the line, (a, b, c) is a direction vector, and λ is a parameter.

The distance from a point (x₀, y₀, z₀) to a plane ax + by + cz + d = 0 is calculated as |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²).

Two planes in 3D space are parallel if they have the same normal vector.

The vector (a, b, c) in the equation ax + by + cz + d = 0 represents the normal vector to the plane.

Two lines in 3D space either intersect or are skew. If they intersect, they will have a common point in space.

A plane in 3D space can be represented by an equation of the form ax + by + cz + d = 0, where a, b, and c are not all zero.

Two lines that are neither parallel nor intersecting.

A plane in 3D space is defined by the equation ax + by + cz + d = 0, where a, b, c, and d are constants.

A line in 3D space is represented by Q(λ)=(x0+a1λ, y0+a2λ, z0+a3λ), where x0, y0, z0 are coordinates of a point on the line, and a1, a2, a3 are direction numbers, and λ is a parameter.

A line and a plane in 3D space can intersect at one point or not intersect at all, or the line is on the plane.

If the line lies entirely within the plane, all points on the line satisfy the plane equation.

The parameter λ̃ (lambda tilde) for the intersection of a line and a plane is calculated as ( - (ax0 + by0 + cz0 + d)) / (aa1+ba2+c*a3).

To find the plane containing two parallel lines, choose a point on each line. Then use one of the given lines and a line constructed through the chosen points to determine the plane.

The intersection point of a line and a plane is valid only if the denominator in the calculation for λ̃ is non-zero.