What a vector is

A vector is a mathematical object with both magnitude and direction. In this topic, vectors usually represent displacement, velocity, or a directed step from one point to another. By contrast, a scalar has magnitude but no direction. Distance and speed are standard examples.

Geometrically, we show a vector as a directed line segment: a line segment with an arrow to indicate direction. The vector $\vec{AB}$ begins at point $A$ and ends at point $B$. You can draw the same vector anywhere on the page, provided its length and direction stay unchanged. Its position doesn’t define it.

A displacement vector gives the change from one position to another. A position vector runs from the origin to a point. If point $P$ has coordinates $(x,y,z)$, its position vector is

where $\vec{OP}$ is the position vector of $P$ measured in coordinate units. The values $x$, $y$ and $z$ are the coordinates of $P$ in coordinate units, while $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are unit base vectors in the positive $x$-, $y$- and $z$-directions respectively.

A general vector can be written in component form as

where $\mathbf{v}$ is a vector measured in coordinate units. Its components $v_1$, $v_2$ and $v_3$ lie in the $\mathbf{i}$-, $\mathbf{j}$- and $\mathbf{k}$-directions respectively. In two dimensions, we simply leave out the third component.

From position vectors to displacement vectors

Suppose $\vec{OA}=\mathbf{a}$ and $\vec{OB}=\mathbf{b}$. The displacement from $A$ to $B$ is then

where $\vec{AB}$ is the displacement vector from $A$ to $B$, $\mathbf{a}$ is the position vector of $A$, and $\mathbf{b}$ is the position vector of $B$. This small fact is used throughout the topic: terminal minus initial.

The distance between points $A$ and $B$ is the magnitude of $\vec{AB}$. It isn’t a new distance formula; it’s Pythagoras written in vector language.

Magnitude and unit vectors

The magnitude of a vector is the non-negative length of the directed line segment that represents it. For

the magnitude is

where $|\mathbf{v}|$ is the length of $\mathbf{v}$ in coordinate units.

A unit vector has magnitude $1$. If $\mathbf{v}$ is non-zero, the unit vector pointing in the same direction is

where $\frac{\mathbf{v}}{|\mathbf{v}|}$ is the vector parallel to $\mathbf{v}$ with length $1$.

Adding, subtracting and scaling vectors

Add vectors component by component:

where $a_1$, $a_2$, $a_3$ are the components of a vector $\mathbf{a}$ and $b_1$, $b_2$, $b_3$ are the components of a vector $\mathbf{b}$.

In a diagram, addition means placing the second vector after the first, head-to-tail. Subtracting a vector is the same as adding its reverse. The zero vector, written $\mathbf{0}$, has magnitude $0$ and no direction. Meanwhile, $-\mathbf{v}$ has the same magnitude as $\mathbf{v}$ but points in the opposite direction, so

A scalar multiple is formed by multiplying a vector by a scalar. If $k$ is a scalar with no direction, then

where $k\mathbf{v}$ is the vector obtained by multiplying each component of $\mathbf{v}$ by $k$. When $k>0$, the direction stays the same. If $k<0$, the vector reverses, and if $k=0$, the result is $\mathbf{0}$.

Two non-zero vectors are parallel vectors when one is a scalar multiple of the other. Thus, $\mathbf{u}$ and $\mathbf{v}$ are parallel exactly when $\mathbf{u}=k\mathbf{v}$ for some scalar $k$, where $\mathbf{u}$ is another vector in coordinate units.

Vector proofs in geometry

Vector geometry proofs work because algebraic equality and geometric equality express the same statement in different forms. To show that two line segments are parallel, prove that their vectors are scalar multiples. For equal directed segments, show that the vectors have equal components. To establish that a quadrilateral has opposite sides equal and parallel, compare the displacement vectors of those sides.

This makes vectors particularly useful in position-location problems: they turn space into algebra. That power isn’t ethically neutral. The same mathematics might guide a rescue vehicle to a lost sailor or direct a destructive device towards a target. Knowledge alone doesn’t decide how it will be used, but the person using it remains responsible for the choices attached to it.

Definition and component form

The scalar product, also called the dot product, takes two vectors and produces a scalar. It measures how much the vectors point in the same direction.

For two non-zero vectors $\mathbf{v}$ and $\mathbf{w}$,

Angles are usually measured in radians unless degrees are stated.

If

the component form is

Combining the two forms gives the angle formula:

Learn to recognise this formula straight away. In two dimensions, the third terms simply disappear.

Perpendicular and parallel vectors

Two non-zero vectors are perpendicular vectors when the angle between them is $\frac{\pi}{2}$ radians. Since $\cos\frac{\pi}{2}=0$,

is equivalent to $\mathbf{v}$ and $\mathbf{w}$ being perpendicular. In this vector setting, orthogonal means perpendicular.

For non-zero parallel vectors, the angle is $0$ or $\pi$. Therefore,

Watch the absolute value here. Parallel vectors can point in the same direction or in opposite directions.

Properties of the scalar product

The scalar product works predictably with addition and scalar multiplication:

where $\mathbf{u}$ is a third vector measured in coordinate units. These aren’t merely algebra rules to memorise. They show that projection-like information is preserved when vectors are split into sums.

Why define this product rather than a different one? It creates an efficient link between geometry and algebra: lengths, angles, perpendicularity and the cosine rule all follow from one definition. The same operation appears in physics whenever only the component of one vector in the direction of another matters, such as a force acting along a displacement.

The vector equation

You can describe a line using one point on it and a direction parallel to it. Its vector equation is

The key phrase here is variable position vector. The vector $\mathbf{r}$ isn't the line itself. As $\lambda$ changes, the endpoint of $\mathbf{r}$ traces out the line.

For a line passing through points $A$ and $B$, use $\vec{AB}=\mathbf{b}-\mathbf{a}$ to find a direction vector, following the notation from section 3.12. Either $\vec{AB}$ or $\vec{BA}$ works; they simply point in opposite directions along the same line.

Parametric and Cartesian forms

Suppose

Then

where $x$, $y$ and $z$ give the coordinates of a general point on the line. Equating the components produces the parametric form:

Provided $l$, $m$ and $n$ are non-zero, eliminate $\lambda$ to obtain the Cartesian form:

Never divide by a direction component that equals $0$. For instance, if $m=0$, keep $y=y_0$ as a separate equation and equate the remaining two fractions. This small detail avoids a lot of nonsense.

These forms are equivalent, though each suits a different purpose. Vector form displays the point and direction clearly. Parametric form is usually easiest for substitution, while Cartesian form is compact when every direction component is non-zero. A mathematician chooses a representation based on clarity and efficiency, as well as how well it supports the next calculation.

The angle between two lines

The angle between two lines is the acute or obtuse angle between their direction vectors. Suppose the lines have direction vectors $\mathbf{b}_1$ and $\mathbf{b}_2$

Then

To find the acute angle, take the absolute value of the numerator:

Simple kinematics interpretation

For a straight-line motion model, read the same equation as

The constant velocity vector $\mathbf{b}$ is measured in coordinate units per second, and $\mathbf{r}$ gives the position vector at time $t$. In this setting, $|\mathbf{b}|$ is the speed in coordinate units per second.

This is the vector form of constant velocity motion. It also explains why vectors fit navigation systems such as GPS so naturally: position plus time times velocity gives a compact model of linear motion in space.

Four possible relationships

In two dimensions, non-parallel lines meet. That isn’t always the case in three dimensions, so spatial awareness matters here.

Two lines are coincident lines when they contain exactly the same set of points. Their direction vectors are parallel, and they share at least one point.

Two lines are parallel distinct lines when their direction vectors are scalar multiples but the lines have no point in common.

Two lines are intersecting lines if they share exactly one point. Their direction vectors are not scalar multiples, and solving the two line equations produces a consistent pair of parameter values.

Two lines are skew lines when they are non-parallel and do not intersect in three-dimensional space. This is a genuinely three-dimensional case: skew lines cannot lie in a single flat plane.

How to distinguish the cases

Suppose

and

Start by checking whether $\mathbf{b}_1$ and $\mathbf{b}_2$ are parallel. If they are, take a vector joining a point on one line to a point on the other and check whether it is parallel to the same direction. In algebraic terms, test whether $\mathbf{a}_2-\mathbf{a}_1$ is a scalar multiple of $\mathbf{b}_1$.

When the direction vectors aren’t parallel, set the vector equations equal:

Equating components produces three scalar equations for the two unknown parameters $\lambda$ and $\mu$. A consistent system means that the lines intersect. Substitute the parameter value into either line to find the point of intersection. If the system is inconsistent, the lines are skew.

Here, geometry turns into a system of linear equations. Coincident lines have infinitely many points of intersection, although a calculation may describe all of them using one parameter. Mathematics represents an infinite solution set by giving the structure that generates it rather than listing every solution.

Definition and direction

The vector product, also called the cross product, takes two vectors in three dimensions and produces a vector perpendicular to both.

For two non-zero vectors $\mathbf{v}$ and $\mathbf{w}$,

Its direction comes from the right-hand screw rule.

The right-hand screw rule matters. Curl the fingers of your right hand from the first vector towards the second. Your thumb then points in the direction of $\mathbf{v}\times\mathbf{w}$. If you reverse the order, the direction reverses too.

Component form

For

the component form is

Write this carefully as a vertical column. Most incorrect answers here come from sign errors or swapped terms rather than a failure to understand the concept.

Properties of the vector product

The vector product is not commutative:

It distributes over addition:

A scalar can be taken outside:

Crossing a vector with itself gives the zero vector:

For non-zero vectors,

is equivalent to $\mathbf{v}$ and $\mathbf{w}$ being parallel. This gives the cross-product test for parallel vectors.

Area interpretation

Taking magnitudes in the definition gives

This is base multiplied by perpendicular height. Therefore, $|\mathbf{v}\times\mathbf{w}|$ is the area of the parallelogram whose adjacent sides are $\mathbf{v}$ and $\mathbf{w}$. The corresponding triangle has area

What defines a plane

A plane is a flat two-dimensional surface that extends indefinitely through three-dimensional space. A line needs one direction. A plane needs two independent directions, or one normal direction.

There are several ways to define one: three non-collinear points; one point with two non-parallel direction vectors; or one point with a normal vector. In a question, choose the form that fits the given information most directly.

Parametric vector form

The parametric vector equation of a plane is

Equating the $\mathbf{i}$-, $\mathbf{j}$- and $\mathbf{k}$-components produces three parametric equations. If the problem calls for a Cartesian equation, eliminate the two parameters.

Normal form and Cartesian form

A normal vector is perpendicular to every direction vector lying in a plane. If $\mathbf{n}$ is a normal vector to a plane and $\mathbf{a}$ is the position vector of a point on that plane, then

where $\mathbf{n}$ is measured in coordinate units and is perpendicular to the plane.

The Cartesian equation of a plane is

Don’t confuse this with the equation of a line. In two dimensions, $Ax+By=D$ is a line; in three dimensions, $Ax+By+Cz=D$ is a plane. A Cartesian equation of a line in three dimensions normally requires two equalities or two plane equations together.

Finding a normal vector

Suppose a plane contains two non-parallel direction vectors $\mathbf{b}$ and $\mathbf{c}$. Then

is a normal vector to the plane. Once you know $\mathbf{n}$ and one point on the plane, use $\mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n}$ or substitute into $Ax+By+Cz=D$.

Here, the algebraic, geometric and vector representations connect directly. The cross product gives a perpendicular direction. The dot product then uses that perpendicularity to fix the plane in place, and the Cartesian form presents the result in a convenient form for substitution.

Intersections as systems of equations

In three dimensions, intersections are usually found by substitution or by solving simultaneous linear equations. The geometry helps you predict the kind of answer you should get.

A line and a plane can meet at one point, be parallel with no intersection, or have the whole line contained in the plane. Substitute the line equation into the plane equation to test which case applies. One value of the line parameter gives one intersection point. A contradiction means there is no intersection, while an identity shows that the line lies in the plane.

Two non-parallel planes meet in a line. Parallel planes are either distinct, so they have no intersection, or coincident, with infinitely many shared points. Three planes might meet at one point, along a line, in a plane, or have no common point. Algebra gives the same possibilities as a system of equations: a unique solution, infinitely many solutions, or inconsistency.

Here is the connection to linear systems. Vector language describes the geometry, but component equations handle the solving. The key skill is moving confidently between the two representations.

Angle between two planes

The absolute value gives the acute angle directly. For a Cartesian plane equation, read the normal vector straight from the coefficients of $x$, $y$ and $z$.

Angle between a line and a plane

The angle between a line and a plane is not the angle between the line's direction and the plane's normal. Instead, it is the complement of that angle.

This sine formula usually gives the line-plane angle most directly. Alternatively, first use cosine to find the angle between $\mathbf{b}$ and $\mathbf{n}$, then subtract it from $\frac{\pi}{2}$.

A practical route through mixed problems

When lines and planes appear in the same problem, first identify what each equation tells you. A vector line supplies a point and a direction. A Cartesian plane immediately supplies a normal vector, while a parametric plane gives a point and two directions. Then pick the operation suited to the task: use the dot product for angles and perpendicularity, the cross product for normals and areas, and substitution for intersections.

These methods model three-dimensional relationships in flight paths, navigation, scanning devices and line-of-sight calculations. One representation may make a mathematical object easy to use while another makes it awkward, so choose the form that reveals the relationship needed for the next step.