Difficulty: Intermediate | Prerequisites: Single-variable calculus (derivatives, integrals), basic trigonometry, familiarity with 2D coordinate geometry.
This is the first half of a Calculus III (multivariable calculus) course, covering the foundational geometry of three-dimensional space and the algebra of vectors. Everything here underpins the later topics of partial derivatives, multiple integrals, and vector calculus. If you can differentiate and integrate single-variable functions and you are comfortable with 2D coordinate planes, you are ready for this material. If you skipped the last few weeks, start here: vectors and their operations are the language the rest of the course is written in.
TL;DR
Vectors in 3D space have magnitude and direction and can be added, scaled, and multiplied (dot product for angles and work, cross product for perpendicularity and area). Lines and planes in space each have standard equation forms built from points and direction or normal vectors. Distance formulas extend naturally from 2D to 3D.
Vector
A quantity with both magnitude (length) and direction, represented as a directed line segment or an ordered triple of components. In simple terms, a vector tells you "how far and which way."
Scalar
A plain number (no direction), such as length, temperature, or mass. Think of it as the magnitude without a heading.
Unit vector
A vector with magnitude 1, used to indicate direction only. You get the unit vector of any non-zero vector by dividing it by its own magnitude.
Position vector
The vector from the origin to a point P, written OP = (x, y, z). Think of it as the address of a point in space.
Displacement vector
The vector from one point to another, computed as the difference of their position vectors: AB = B - A. In simple terms, this is the "shift" that takes you from A to B.
Dot product (scalar product)
A way of multiplying two vectors that produces a scalar: a . b = a1b1 + a2b2 + a3*b3. Think of it as a measure of how much two vectors point in the same direction.
Cross product (vector product)
A way of multiplying two vectors that produces a new vector perpendicular to both: a x b. In simple terms, the cross product gives you a vector that sticks straight out of the plane the original two lie in.
Scalar projection
The signed length of one vector's shadow on another. It tells you how far b reaches in the direction of a.
Vector projection
The actual vector (not just its length) that represents the shadow of b onto a.
Normal vector
A vector perpendicular to a plane. Any plane in 3D space can be defined by a single point on it plus a normal vector.
Direction vector
A vector parallel to a line, used in the parametric or vector equation of that line.
Octant
One of the eight regions that the three coordinate planes divide 3D space into. The first octant is where x, y, and z are all positive.
Orthogonal (perpendicular)
Two vectors are orthogonal when their dot product equals zero. In simple terms, they meet at a right angle.
Parallel vectors
Two vectors are parallel when one is a scalar multiple of the other. Their cross product is the zero vector.
Points in space are located by ordered triples (x, y, z) relative to three mutually perpendicular axes.
The three coordinate planes (xy-plane where z = 0, xz-plane where y = 0, yz-plane where x = 0) divide space into eight octants.
The first octant is the region where all three coordinates are positive.
Visualisation trick: stand in a room corner. The floor is the xy-plane, one wall is the xz-plane, the other wall is the yz-plane, and the ceiling extends the z-axis upwards.
A vector has both magnitude and direction. It is drawn as a directed line segment (an arrow).
Notation: boldface v, or with an arrow above the letter.
Component form: any vector a in 3D can be written a = (a1, a2, a3).
Addition: tip-to-tail geometrically, or add corresponding components: (a1 + b1, a2 + b2, a3 + b3).
Scalar multiplication: multiply each component by the scalar. If c > 0 the direction is preserved; if c < 0 it reverses.
Subtraction: a - b = a + (-b); subtract corresponding components.
Magnitude (length): |a| = sqrt(a1² + a2² + a3²).
Unit vector: a/|a|, pointing in the same direction with length 1. The standard unit vectors are i = (1,0,0), j = (0,1,0), k = (0,0,1).
The position vector of a point P = (x, y, z) is the vector from the origin O to P: OP = (x, y, z).
The displacement vector from A to B is AB = B - A, computed by subtracting corresponding coordinates.
Displacement vectors are what let you describe motion and direction between any two points, not just from the origin.
Algebraic definition: for a = (a1, a2, a3) and b = (b1, b2, b3), the dot product is a . b = a1b1 + a2b2 + a3*b3. The result is a scalar, not a vector.
Geometric definition: a . b = |a| |b| cos(theta), where theta is the angle between the two vectors.
Rearranging the geometric form gives the angle: theta = arccos(a . b / (|a| |b|)).
Orthogonality test: two non-zero vectors are perpendicular exactly when a . b = 0.
Key properties: commutative (a . b = b . a), distributive over addition, and a . a = |a|².
Scalar projection of b onto a: comp_a b = (a . b) / |a|. This gives the signed length of the "shadow" of b on the line through a.
Vector projection of b onto a: proj_a b = ((a . b) / |a|²) a. This gives the actual vector component of b along a.
Physical picture: if you shine a light straight down onto the line of a, the shadow of b on that line is the vector projection.
Work as a dot product: when a constant force F moves an object along a displacement D, the work done is W = F . D. Only the component of force in the direction of motion contributes to work.
Definition: for a = (a1, a2, a3) and b = (b1, b2, b3), the cross product is a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).
The result is a vector, not a scalar.
Perpendicularity: a x b is perpendicular to both a and b.
Direction: determined by the right-hand rule. Curl the fingers of your right hand from a toward b; your thumb points in the direction of a x b.
Magnitude: |a x b| = |a| |b| sin(theta). This equals the area of the parallelogram formed by a and b.
Anti-commutative: b x a = -(a x b). Order matters.
Parallel test: if a x b = 0, then a and b are parallel (or one is the zero vector).
Applications: finding a normal vector to a plane defined by two vectors, computing torque, and calculating areas of parallelograms and triangles (triangle area = half the parallelogram area).
Lines in 3D
A line is defined by a point on it and a direction vector v = (a, b, c).
Vector form: r(t) = r_0 + t v, where r_0 is the position vector of a known point.
Parametric form: x = x_0 + at, y = y_0 + bt, z = z_0 + ct.
Symmetric form: (x - x_0)/a = (y - y_0)/b = (z - z_0)/c, valid when a, b, c are all non-zero.
Two lines in 3D that are not parallel and do not intersect are called skew lines.
Planes in 3D
A plane is defined by a point on it and a normal vector n = (a, b, c) perpendicular to the plane.
Equation: a(x - x_0) + b(y - y_0) + c(z - z_0) = 0, which simplifies to ax + by + cz = d.
To find the normal to a plane containing two vectors u and v, take n = u x v.
Two planes are parallel if their normal vectors are parallel; perpendicular if their normals have a dot product of zero.
Point to point: d = sqrt((x2 - x1)² + (y2 - y1)² + (z2 - z1)²). This is the 3D extension of the Pythagorean theorem.
Point to plane: for a plane ax + by + cz + d = 0 and a point (x1, y1, z1), the distance is |ax1 + by1 + c*z1 + d| / sqrt(a² + b² + c²).
Between skew lines: find the perpendicular distance using the cross product of the two direction vectors and the vector connecting a point on each line.
Formula | Expression |
|---|---|
Magnitude of a vector | |
Dot product | a . b = a1b1 + a2b2 + a3*b3 |
Dot product (geometric) | a . b = |
Angle between vectors | theta = arccos(a . b / ( |
Scalar projection | comp_a b = (a . b) / |
Vector projection | proj_a b = ((a . b) / |
Work | W = F . D |
Cross product | a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1) |
Cross product magnitude | |
Parametric line | x = x0 + at, y = y0 + bt, z = z0 + ct |
Plane equation | a(x - x0) + b(y - y0) + c(z - z0) = 0 |
Point-to-point distance | sqrt((x2-x1)² + (y2-y1)² + (z2-z1)²) |
Point-to-plane distance |
Work and energy: computing the work a force does on an object is a direct application of the dot product. This is how engineers size motors, analyse structural loads, and calculate energy expenditure.
Torque: the cross product defines torque (the rotational force on a wrench or bolt). The magnitude depends on the sine of the angle, so maximum torque occurs when the force is perpendicular to the lever arm.
Navigation and computer graphics: planes, lines, and normal vectors underpin how 3D rendering engines determine visibility, lighting, and collision detection.
Students often think the dot product produces a vector. It does not. The dot product always returns a scalar.
Students confuse the direction of the cross product. a x b and b x a point in opposite directions. The cross product is anti-commutative.
A common mistake is to assume two lines in 3D that are not parallel must intersect. In 3D, non-parallel lines can be skew (they never meet and are not in the same plane).
Students sometimes forget that the symmetric equation of a line is undefined when one of the direction components is zero. If, say, b = 0, you write y = y_0 as a separate equation instead.
⚠️ The dot product and cross product are tested heavily. Know which gives a scalar and which gives a vector.
⚠️ Expect exam questions that ask you to find the equation of a plane given three points (find two vectors in the plane, cross them for the normal, plug in).
⚠️ Distance from a point to a plane is a classic exam problem. Memorise the formula and practise applying it.
⚠️ Projection problems frequently appear as word problems involving work or force decomposition.
True or false: The dot product of two vectors is always a vector. (False, it is a scalar.)
True or false: If a x b = 0, then a and b are perpendicular. (False, they are parallel.)
Fill in the blank: The first octant is the region where x ___, y ___, and z ___. (> 0, > 0, > 0)
True or false: Two lines in 3D space that do not intersect must be parallel. (False, they could be skew.)
Fill in the blank: The vector projection of b onto a points in the direction of ___. (a)
Q: Find the angle between a = (1, 2, 3) and b = (4, -1, 2).
A: Compute a . b = 4 - 2 + 6 = 8. Then |a| = sqrt(14), |b| = sqrt(21). So theta = arccos(8 / sqrt(294)) which is approximately 62.2 degrees.
Q: Find a vector perpendicular to both u = (1, 0, 2) and v = (3, 1, -1).
A: Compute u x v = (0*(-1) - 21, 23 - 1*(-1), 11 - 03) = (-2, 7, 1).
Q: Find the equation of the plane through the point (1, 2, 3) with normal vector n = (4, 5, 6).
A: 4(x - 1) + 5(y - 2) + 6(z - 3) = 0, which simplifies to 4x + 5y + 6z = 32.
Q: Compute the work done by force F = (3, -1, 2) along displacement D = (1, 4, -2).
A: W = F . D = 3 - 4 - 4 = -5. The negative value means the force opposes the motion (on net).
Q: Find the distance from the point (2, 1, -1) to the plane 3x - y + 2z = 5.
A: Rewrite as 3x - y + 2z - 5 = 0. Distance = |3(2) - 1(1) + 2(-1) - 5| / sqrt(9 + 1 + 4) = |6 - 1 - 2 - 5| / sqrt(14) = 2/sqrt(14) which is approximately 0.535.
Connections to other topics
This material connects directly to surfaces and quadric surfaces (you will use planes and normal vectors to describe tangent planes). It also connects to space curves, where the cross product is used to compute curvature and the TNB frame. The dot product reappears in directional derivatives and the gradient vector. If you are taking physics concurrently, vectors and projections are the same tools used for force decomposition, work-energy theorems, and torque.
Related Terms / Search Tags
vectors in 3D, dot product, scalar product, cross product, vector product, vector addition, scalar multiplication, unit vector, position vector, displacement vector, orthogonal vectors, perpendicular vectors, parallel vectors, right-hand rule, parametric equations of a line, symmetric equations, equation of a plane, normal vector, direction vector, scalar projection, vector projection, work as dot product, distance point to plane, distance between skew lines, octant, coordinate planes, Calculus III, MATH 215, multivariable calculus, geometry in space