A vector in three dimensions is a mathematical object that has both magnitude and direction. It is represented by a line segment with an arrow pointing in the direction of the vector. In three dimensions, the vector extends in three different directions (x, y, and z axes) and is often denoted by v.
A vector in three dimensions is typically represented using coordinates, also known as components. These coordinates can be written as an ordered triple (x,y,z) or as v = xi + yj + zk, where i, j, and k are unit vectors along the x, y, and z axes, respectively.
The magnitude of a vector in three dimensions is the length of the vector. It can be calculated using the Pythagorean theorem, where the magnitude of the vector is the square root of the sum of the squares of its components: |v| = √(x^2 + y^2 + z^2).
To add or subtract two vectors in three dimensions, you can use the parallelogram rule or the head-to-tail method. In the parallelogram rule, you draw a parallelogram with the two vectors as adjacent sides, and the diagonal represents the sum or difference of the two vectors. In the head-to-tail method, you place the tail of one vector at the head of the other vector, and the sum or difference is the vector from the tail of the first vector to the head of the second vector.
The dot product of two vectors in three dimensions is a scalar quantity that represents the projection of one vector onto the other. It is calculated by multiplying the corresponding components of the two vectors and adding them together: v · w = (xv * xw) + (yv * yw) + (zv * zw). The cross product, on the other hand, is a vector quantity that is perpendicular to both vectors and has a magnitude equal to the product of their magnitudes and the sine of the angle between them: v x w = |v| * |w| * sin(θ).