MATH1061 · Mathematics 1a
Vectors
A vector is an ordered list of numbers — geometrically an arrow you can add, scale and measure — and it is the language of every later linear-algebra topic. The single most powerful tool is the dot product, u·v = ‖u‖‖v‖cosθ, which is one scalar that encodes the angle between two vectors, decides orthogonality (dot product zero), and produces the projection of one vector onto another. In ℝ³ the cross product u × v gives a vector perpendicular to both (right-hand rule) whose length is the area of the parallelogram they span. These combine into the geometry of lines and planes in ℝ³: a line is a point plus a direction (parametric form), a plane is a point plus a normal (normal form), and the dot/cross products deliver the distances and angles between them.
What this chapter covers
- 01B4 Vector arithmetic, length and the parallelogram law
- 02B5 The dot product, angle and orthogonality
- 03B5 Projection of one vector onto another
- 04B6 The cross product in ℝ³ — the right-hand rule, parallelogram area
- 05B7 Lines in ℝ³ — vector / parametric form
- 06B7 Planes in ℝ³ — normal form, distances and angles
Worked example: angle and cross product in ℝ³
- +1(a) Dot product: u·v = (1)(2) + (2)(0) + (2)(−1) = 2 + 0 − 2 = 0.
- +1(a) Interpret: u·v = 0 means the vectors are orthogonal — the angle is exactly 90°.
- +1(a) Confirm via the formula: cosθ = u·v / (‖u‖‖v‖) = 0 / (3·√5) = 0, so θ = 90°.
- +1(b) Set up the cross product: a vector perpendicular to both is u × v.
- +1(b) Compute: u × v = (2·(−1) − 2·0, 2·2 − 1·(−1), 1·0 − 2·2) = (−2, 5, −4).
- +1(b) Check: (−2, 5, −4)·u = −2 + 10 − 8 = 0 and ·v = −4 + 0 + 4 = 0 — perpendicular to both, as required.
Key terms
- Dot (scalar) product
- u·v = u₁v₁ + u₂v₂ + u₃v₃ = ‖u‖‖v‖cos θ — a single number encoding the angle. Zero dot product means the vectors are orthogonal (perpendicular).
- Norm (length)
- ‖u‖ = √(u·u) = √(u₁² + u₂² + u₃²), the length of the vector. A unit vector has norm 1; dividing a vector by its norm normalises it.
- Projection
- The projection of v onto u is (u·v / ‖u‖²) u — the shadow of v in u's direction. It splits v into a component along u and a component perpendicular to u.
- Cross product
- In ℝ³, u × v is the vector perpendicular to both u and v (direction by the right-hand rule) with length ‖u‖‖v‖sin θ = the area of the parallelogram they span. It is zero exactly when u and v are parallel.
- Plane (normal form)
- A plane in ℝ³ is fixed by a point and a normal vector n: every point r on it satisfies n·(r − r₀) = 0, i.e. n·r = d. The normal's components are the coefficients of x, y, z in the Cartesian equation.
Vectors FAQ
What does the dot product tell me?
Three things at once. Its sign and value, via u·v = ‖u‖‖v‖cos θ, give the angle between the vectors. A dot product of exactly zero means they're orthogonal (perpendicular). And the dot product drives the projection formula. Whenever a question asks about an angle, perpendicularity, or a shadow/component, reach for the dot product first.
When do I use the cross product instead of the dot product?
Use the cross product when you need a vector perpendicular to two others (a normal to a plane), or an area/volume. It exists only in ℝ³ and returns a vector; the dot product works in any dimension and returns a scalar. Mnemonic: dot → angle/scalar, cross → perpendicular vector/area.
How do I write the equation of a line versus a plane in ℝ³?
A line needs a point and a direction vector: r = r₀ + t·d (parametric form). A plane needs a point and a normal vector: n·(r − r₀) = 0, which expands to the Cartesian ax + by + cz = d where (a, b, c) is the normal. The recurring move is finding the right normal — often via a cross product of two vectors lying in the plane.
How do I find the distance from a point to a plane?
Take the vector from any point on the plane to the external point, then project it onto the unit normal: the distance is |n·(p − r₀)| / ‖n‖. The dot product with the normalised normal extracts exactly the perpendicular component, which is the shortest distance.
Exam move
Make the dot product your default tool: it answers angle, orthogonality and projection questions in one formula, so compute u·v early and read off what the question needs. Reserve the cross product for ℝ³ perpendicular-vector, normal and area problems, and always verify your result with a dot-product check (it must be 0 against both inputs). For lines and planes, fix the form first — point + direction for a line, point + normal for a plane — then translate to parametric or Cartesian. Keep components and surds exact, and draw a quick sketch for sign and right-hand-rule sanity. Method marks come from showing the formula, not just the final number.
