3D vectors are mathematical objects used to represent quantities with both magnitude (length) and direction in a three-dimensional space. They consist of three components: x, y, and z, and are often represented as (x, y, z).
The magnitude of a 3D vector can be found using the Pythagorean theorem, which states that the magnitude is equal to the square root of the sum of the squares of each component. In other words, the magnitude of a 3D vector (x, y, z) is equal to √(x² + y² + z²).
A vector represents a direction and magnitude in space, while a point represents a specific location in space. Vectors are often used to describe the relationship between points and can be used to calculate distances and angles between points.
To determine if a point (x, y, z) lies on a line, you can use the parametric equation of a line, which is given by x = x₀ + at, y = y₀ + bt, and z = z₀ + ct, where (x₀, y₀, z₀) is a point on the line and a, b, and c are the direction ratios. If plugging in the coordinates of the point into this equation results in a true statement, then the point lies on the line.
The cross product of two 3D vectors is a vector that is perpendicular to both of the original vectors. It is calculated using the formula (a, b, c) × (d, e, f) = (bf - ce, cd - af, ae - bd). The magnitude of this cross product is equal to the product of the magnitudes of the two original vectors multiplied by the sine of the angle between them.