Vectors
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Vector arithmetic & geometry
2D vector notation with base vectors
The equation of a vector in two dimensions can be written as a sum of its vertical and horizontal components, denoted i and j respectively. These are vectors that start from the origin and point right (i) and up (j), each with a length of 1. We call these base vectors.
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2D vector notation with column vectors
A vector in two dimensions can be uniquely described by giving its vertical and horizontal components, meaning the number of units in each direction required to get from the "tail" to the "head" of the arrow. These components can be represented using a column vector.
The column vector of a vector v whose head is located a horizontal and b vertical units from its tail is
The top number represents the horizontal component and the bottom number represents the vertical component.
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3D base and column vectors
A vector is a mathematical object that has both magnitude (size) and direction in space. A three-dimensional vector is typically described by its components along the x, y, and z axes.
In 3 dimensions, a vector is expressed as
where i, j, and k are called the base vectors:
Adding & subtracting vectors
To add two vectors, add their corresponding components:
To subtract two vectors, subtract their corresponding components:
Addition corresponds to placing vectors head-to-tail, and subtraction corresponds to the displacement between their tips.
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Position & Displacement Vectors
A position vector describes the position of a point relative to a fixed origin. For example, the position vector of a point P describes the location of P relative to an origin O:
A displacement vector describes how to move from one point to another. For instance, the displacement vector from point A to point B is given by subtracting position vectors:
Thus, position vectors identify points relative to an origin, while displacement vectors represent movement or translation between points. Expressing AB as OB−OA is sometimes referred to as giving the relative position of B from A.
Zero vector and negative vector
The zero vector 0 is a special vector without size or a defined direction, represented by
The negative of a vector reverses the vector's direction while maintaining its size. If v=⎝⎛xyz⎠⎞, then the negative is
pointing in exactly the opposite direction.
Scalar multiples of vectors
Scaling a vector means multiplying it by a number k, which changes its length without affecting its direction (unless k<0, which reverses direction). If
then scaling by k gives
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Parallel vectors
Two vectors are parallel if one can be written as a scalar multiple of the other, i.e., if u=kv for some scalar k. Parallel vectors have identical or exactly opposite directions.
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Vector Magnitude
The magnitude (or length) of a vector v=⎝⎛v1v2v3⎠⎞ is calculated as
Geometrically, this represents the distance from the vector's starting point to its endpoint in 3D space. For example, the magnitude of the displacement vector AB equals the straight-line distance between points A and B.
Unit vectors
When a vector u has a magnitude of 1, we say u is a unit vector.
We often express other vectors v with magnitude ∣v∣=1 as scalar multiples of the unit vector pointing in the same direction,
where k=∣v∣.
The unit vector in the same direction as a non-unit vector v is written as
Rescaling to form a vector with a magnitude of one is referred to as normalizing a vector.
Scalar and vector products
Scalar product
The scalar product, also called the dot product, takes two vectors and produces a scalar (a number). For two vectors v=⎝⎛v1v2v3⎠⎞ and w=⎝⎛w1w2w3⎠⎞, the scalar product is calculated as:
This operation combines corresponding components of each vector, resulting in a single numerical value.
Angle between vectors
The scalar product also has a geometric interpretation involving the angle θ between two vectors:
Equivalently, isolating cosθ:
The angle θ is measured between the heads of v and w:
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Acute vs Obtuse Angles
If the scalar product of two vectors is negative, then
and thus θ must be an obtuse angle: 90°<θ≤180°.
But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting
whenever θ>90°.
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Perpendicular vectors
Since v⋅w=∣v∣∣w∣cosθ, if two vectors are perpendicular then cos90°=0⇒ their scalar product is 0.
Properties of the scalar product
For any vectors u, v, and w, and scalar k:
Vector product
The vector product, sometimes called the cross product, of two vectors v and w is given by:
This vector is perpendicular to both original vectors. Using the right-hand rule, if your index finger points in the direction of v and your middle finger points towards w, your thumb points towards v×w. The vector product thus creates a new vector perpendicular to both original vectors.
Vector areas with cross product
The magnitude of the vector product v×w gives the area of the parallelogram formed by vectors v and w:
Intuitively, this happens because the magnitude combines both vectors’ lengths and how "spread out" they are from each other, capturing exactly the amount of two-dimensional space they span.
Vector product and sin of angle
The magnitude of the vector product is connected to the sine of the angle between the vectors by the formula
Here, θ is the angle between vectors v and w. This relationship holds because the area of the parallelogram formed by the two vectors depends on their lengths and the angle separating them.
Specifically, the area is largest when the vectors are perpendicular (sin90∘=1) and zero when they are parallel (sin0∘=0).
Components of vectors
For two vectors a and b, the magnitude of the component of vector a that acts in the direction of vector b is given by
The magnitude of the component of vector a that acts perpendicular to vector b, in the plane formed by the two vectors, is given by
where θ is the angle between the two vectors.
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Properties of the vector product
For any vectors u, v, and w, and scalar k:
Equations of a line
Vector form
A vector line is defined by specifying one fixed point on the line and a direction vector. If the fixed point is given by position vector a, and the direction vector by b, then the line in vector form is:
Here, λ∈R is a parameter that varies over all real numbers, generating every point on the line. Changing λ moves the point r along the direction of b, creating the full infinite line.
Parametric form
A vector line in three-dimensional space can also be expressed in parametric form, showing explicitly the equations for the x, y, and z coordinates separately:
Here, (x0,y0,z0) is a point on the line, (l,m,n) are the components of the direction vector, and the parameter λ controls your position along the line. Changing λ moves the point continuously along the direction of the vector, producing the full infinite line.
Cartesian form
The Cartesian form of a vector line in 3D is obtained by eliminating the parameter λ from the parametric form.
Solving each equation for λ gives:
Equating these expressions yields the Cartesian form:
which clearly emphasizes the consistent ratio of coordinate changes along the line.
If one of the direction vector components is zero, then that coordinate does not change as you move along the line.
Modeling with vectors
In kinematics (the mathematical description of motion), 3D motion can be modeled by a vector line, often expressed with the parameter representing time, t.
Specifically, the direction vector b represents the velocity of an object, indicating both its direction and magnitude of movement. The magnitude of this vector, ∣b∣, is the object's speed—the rate at which it moves, irrespective of direction.
Angle between lines
The angle between two lines is simply the angle between their direction vectors.
For any two lines r1=a1+λb1 and r2=a2+μb2, the angle θ between r1 and r2 can be found via the formula
which is just the equation of the scalar product.
Coincident, Parallel, Intersecting & Skew Lines
Parallel lines in 3D
Two vector lines are parallel if their direction vectors are scalar multiples of each other and the lines are not the same.
Consider two lines:
These lines are parallel if b1=kb2 for some scalar k, but a1 does not lie on r2.
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Coincident lines
Two vector lines are coincident if they represent exactly the same line, meaning every point on one line also lies on the other. For lines given by:
they are coincident if:
Their direction vectors are parallel, so b1=kb2.
A point from one line (e.g., a2) lies on the other line, satisfying a2=a1+λb1 for some scalar λ.
Intersecting lines
Two vector lines intersect if they share exactly one common point. That means they are not parallel.
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Skew lines
Two lines in three-dimensional space are skew if they are neither parallel nor intersecting.
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Equations of a plane
Vector form
A plane in 3D space can be described by a vector equation involving a fixed point and two direction vectors lying in the plane. Planes are often denoted by Π (capital pi).
If the position vector of the fixed point is a, and two non-parallel direction vectors in the plane are b and c, then the plane is represented by:
Here, λ and μ are parameters that can take any real values, allowing r to move freely across the entire surface of the plane.
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Scalar product form
The scalar product form of a plane uses a vector perpendicular ("normal") to the plane and one known point in the plane. If a point in the plane has position vector a and n is a normal vector, then any other point r lies in the plane if:
This equation expresses the idea that the vector from the known point to any other point in the plane is always perpendicular to n.
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Cartesian equation of a plane
The Cartesian equation of a plane with a normal vector n and containing a point with position vector a is
where n=⎝⎛n1n2n3⎠⎞,d=a⋅n.
Angles and intersections with planes
Intersection of line and plane
To find the intersection of a line and a plane, substitute the vector line equation into the plane equation and solve for the parameter λ.
Angle between line and plane
The angle ϕ between a line and a plane is measured as the complement of the angle between the line’s direction vector d and the plane’s normal vector n.
Since the normal forms an angle of 90° with the plane, we can construct a right angled triangle using the intersection of the normal, the line, and the plane:
If θ is the angle between d and n, then ϕ=180°−90∘−θ=90°−θ
In practice, you can compute θ using cosθ=∣d∣∣n∣d⋅n.
Intersection between 2 planes
The intersection of 2 planes - if they are not parallel - is a line. We can find the line intersection by solving a system of equations.
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Angle between 2 planes
The angle θ between two planes can be found by computing the angle between the two plane normals,
Intersection of 3 planes
When three planes are considered together, their intersection in 3D space can take several forms:
A plane if all three planes coincide.
A single point if the three planes intersect uniquely, meaning their normal vectors are not all parallel or in some degenerate arrangement, and the system of equations has exactly one solution.
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A line if two planes intersect in a line and the third plane also contains that line (or if each pair of planes meets along the same line).
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No intersection if the system of equations is inconsistent (e.g., the planes are arranged in parallel or partially parallel ways that do not share a common point).
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