01 Vectors and Geometry in Space

Representing and Combining Vectors

A point identifies a location, while a vector describes a directed quantity. Common vector quantities include displacement, velocity, acceleration, and force. Two vectors are equal when they have the same magnitude and direction, even if they are drawn at different locations.

In three-dimensional space, a vector is written in component form as

where $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are unit vectors in the positive coordinate directions.

If $P=(x_1,y_1,z_1)$ and $Q=(x_2,y_2,z_2)$, the vector from $P$ to $Q$ is found by subtracting coordinates:

The position vector of $P=(x,y,z)$ is the vector from the origin to $P$:

For vectors with matching dimensions, addition, subtraction, and scalar multiplication are performed componentwise. For example, if $\mathbf{u}=\langle 2,-1,3\rangle$ and $\mathbf{v}=\langle -1,4,2\rangle$, then

and

Geometrically, vector addition follows the head-to-tail rule: place the initial point of the second vector at the terminal point of the first. The sum runs from the initial point of the first vector to the terminal point of the second.

The magnitude of a vector in space is

For a nonzero vector, divide by its magnitude to obtain a unit vector in the same direction:

Takeaway: Components encode a vector algebraically, while magnitude and direction explain its geometric meaning.

Angles, Perpendicularity, and Projection

The dot product produces a scalar from two vectors. For $\mathbf{u}=\langle u_1,u_2,u_3\rangle$ and $\mathbf{v}=\langle v_1,v_2,v_3\rangle$,

For nonzero vectors, the same quantity can be expressed using the angle $\theta$ between them:

Therefore,

The sign gives an immediate geometric test:

• If $\mathbf{u}\cdot\mathbf{v}>0$ and the vectors are nonzero, their angle is acute.

• If $\mathbf{u}\cdot\mathbf{v}=0$, the vectors are perpendicular.

• If $\mathbf{u}\cdot\mathbf{v}<0$, their angle is obtuse.

For example, let $\mathbf{u}=\langle 1,2,-1\rangle$ and $\mathbf{v}=\langle 2,-1,3\rangle$. Then

Because the result is negative, the angle between these vectors is obtuse.

The dot product also measures how much one vector points in another vector's direction. The scalar projection of $\mathbf{u}$ onto $\mathbf{v}$ is

and the vector projection is

In an application such as work by a constant force, the work is the dot product of force and displacement:

Takeaway: Use the dot product for angles, perpendicularity, projections, and directional components.

Perpendicular Directions and Area

The cross product is defined for vectors in $\mathbb{R}^3$. If $\mathbf{u}=\langle u_1,u_2,u_3\rangle$ and $\mathbf{v}=\langle v_1,v_2,v_3\rangle$, then

The result is perpendicular to both input vectors. Its magnitude is

so it equals the area of the parallelogram spanned by the vectors. The area of the triangle formed by the same two vectors is

The direction of the cross product follows the right-hand rule. Curl the fingers of your right hand from $\mathbf{u}$ toward $\mathbf{v}$; your thumb points in the direction of $\mathbf{u}\times\mathbf{v}$. Reversing the order reverses the direction:

For example, if $\mathbf{u}=\langle 1,2,0\rangle$ and $\mathbf{v}=\langle 0,1,3\rangle$, then

Checking with dot products confirms perpendicularity:

and

If two vectors are parallel, their cross product is the zero vector because the angle between them has sine equal to zero.

Takeaway: Use the cross product to construct perpendicular directions and calculate areas in three-dimensional space.

Lines in Space

A line in space is determined by a point and a direction vector. If it passes through $P_0=(x_0,y_0,z_0)$ and has direction vector $\mathbf{v}=\langle a,b,c\rangle$, its vector equation is

where $\mathbf{r}_0=\langle x_0,y_0,z_0\rangle$. In component form, this becomes

When all three direction components are nonzero, the same line can be written in symmetric form:

For example, the line through $(1,-2,3)$ parallel to $\langle 2,1,-4\rangle$ is

or equivalently

To determine whether two parametric lines intersect, set their corresponding $x$-, $y$-, and $z$-coordinates equal and solve for the two parameters. A common solution gives an intersection point. If the direction vectors are scalar multiples, the lines are parallel or coincident. If they are not parallel and no common point exists, the lines are skew.

Takeaway: A point fixes a line's location, and a direction vector fixes how the line extends through space.

Planes and Normal Vectors

A plane is determined by one point and a normal vector. If the plane passes through $P_0=(x_0,y_0,z_0)$ and has normal vector $\mathbf{n}=\langle a,b,c\rangle$, then a point $\langle x,y,z\rangle$ lies in the plane exactly when the displacement from $P_0$ is perpendicular to $\mathbf{n}$:

This gives the point-normal equation

which can be expanded into standard form:

For example, the plane through $P_0=(2,1,-1)$ with normal vector $\langle 3,-2,1\rangle$ satisfies

or

To find a plane through three noncollinear points $P$, $Q$, and $R$, form two vectors in the plane:

Then $\mathbf{u}\times\mathbf{v}$ is a normal vector, which can be substituted into the point-normal equation.

The distance from a point $P=(x_1,y_1,z_1)$ to the plane $ax+by+cz=d$ is

The numerator measures the point's signed displacement relative to the plane, while the denominator accounts for the length of the normal vector.

Takeaway: A plane's equation comes from requiring every point-to-plane displacement to be perpendicular to its normal vector.

A Unified Strategy for Spatial Geometry

Dot and cross products turn geometric relationships into algebraic tests that can be calculated directly.

• Vectors are perpendicular when $\mathbf{u}\cdot\mathbf{v}=0$.

• Vectors are parallel when $\mathbf{u}=k\mathbf{v}$ for some scalar $k$. In $\mathbb{R}^3$, parallel vectors also satisfy $\mathbf{u}\times\mathbf{v}=\mathbf{0}$.

• Three vectors are coplanar when $\left(\mathbf{u}\times\mathbf{v}\right)\cdot\mathbf{w}=0$.

• The area of a parallelogram is $\|\mathbf{u}\times\mathbf{v}\|$.

• The volume of a parallelepiped is $\left|\left(\mathbf{u}\times\mathbf{v}\right)\cdot\mathbf{w}\right|$.

The last expression is the absolute value of the scalar triple product. It combines the base area from a cross product with the component of the third vector perpendicular to that base.

A practical choice of tool is:

1. Use component arithmetic for sums, differences, scalar multiples, and magnitudes.

2. Use the dot product for angles, perpendicularity, projections, and work.

3. Use the cross product for perpendicular directions, triangle areas, and parallelogram areas.

4. Use a direction vector with a point to construct a line.

5. Use a normal vector with a point to construct a plane.

6. Use a cross product of two in-plane directions to obtain a plane's normal vector.

Together, these methods provide a unified algebraic language for geometry in space.