Vectors in 2D and 3D space: A checklist for H2 math mastery

Vectors in 2D and 3D space: A checklist for H2 math mastery

Vectors application checklist: Lines and planes in 3D space (H2 math) .

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Frequently Asked Questions

Vectors possess both magnitude and direction, whereas scalars only have magnitude. In 2D and 3D space, vectors are represented by components along the respective axes, while scalars are single numerical values.
Vector addition and subtraction involve adding or subtracting corresponding components. For example, if vector A = (a1, a2) and vector B = (b1, b2), then A + B = (a1+b1, a2+b2). This principle extends to 3D vectors.
The dot product of two vectors A and B is calculated as |A||B|cosθ, where θ is the angle between them. Alternatively, it can be calculated as A · B = a1b1 + a2b2 (in 2D) or A · B = a1b1 + a2b2 + a3b3 (in 3D).
The angle θ between two vectors can be found using the formula cosθ = (A · B) / (|A||B|), where A · B is the dot product and |A| and |B| are the magnitudes of the vectors.
The cross product of two vectors A and B results in a vector perpendicular to both A and B. In 3D space, its calculated as A x B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).
Two vectors are parallel if one is a scalar multiple of the other. They are perpendicular if their dot product is zero.
In 2D, a line can be defined as r = a + t*d, where a is a point on the line, d is the direction vector, and t is a scalar parameter. In 3D, the same principle applies, but with 3D vectors.
The equation of a plane in 3D space is given by r · n = a · n, where r is a general point on the plane, n is the normal vector to the plane, and a is a known point on the plane. Alternatively, the equation can be written as ax + by + cz = d.