Why Vectors are Everywhere
Ever wondered how a video game character moves smoothly, or how a GPS tells you the shortest route? Both rely on vectors – simple arrows that carry both direction and size.
In simple words, a vector is an arrow that shows how far and in which direction something moves. Think of it like a push you give a toy car: the strength of the push is the length, and the direction you push it is the arrow's pointing.
Understanding Vectors
A vector is usually written in bold ( v ) or with an arrow on top ( \u2192v ). The two key parts are:
- Magnitude – the length of the arrow, like how strong a wind blows.
- Direction – where the arrow points, like north‑south or up‑down.
In the plane (2‑D), we describe a vector by its components along the x‑axis and y‑axis, written as v = (v_x, v_y). In three dimensions we add a z‑component: v = (v_x, v_y, v_z).
Magnitude (Length) of a Vector
To find the length, use the Pythagorean theorem: |v| = √(v_x² + v_y²) in 2‑D, and add the z‑part for 3‑D. It’s like measuring the diagonal of a rectangle.
Unit Vector
A unit vector has a magnitude of 1. It shows only direction. You get it by dividing the vector by its own length: û = v / |v|. Imagine shrinking a giant arrow down to a tiny stick while keeping its pointing direction.
Vector Addition and Subtraction
Adding vectors is as easy as placing the tail of the second arrow at the head of the first. The resulting arrow runs from the start of the first to the end of the second. Algebraically, just add the components:
u + v = (u_x + v_x, u_y + v_y)
Subtraction works the same way but you flip the second arrow before joining.
Worked Example: Adding Two Vectors
Let u = (3, 4) and v = (‑2, 5). Then
u + v = (3 + (‑2), 4 + 5) = (1, 9). The new arrow points mostly upward because the y‑component grew.
Scalar (Dot) Product
The dot product takes two vectors and returns a single number (a scalar). It tells you how much one vector goes in the direction of the other. Formula:
u · v = u_x v_x + u_y v_y (+ u_z v_z for 3‑D)
Another handy way is using magnitudes and the angle θ between them: u · v = |u| |v| cosθ. If the vectors are perpendicular (right‑angled), the dot product is zero because cos90° = 0.
Worked Example: Dot Product and Angle
Take a = (2, 3) and b = (4, ‑1).
Dot product: a·b = 2·4 + 3·(‑1) = 8 – 3 = 5.
Magnitudes: |a| = √(2²+3²)=√13, |b| = √(4²+ (‑1)²)=√17.
Angle: cosθ = (a·b)/(|a||b|) = 5/(√13·√17). θ ≈ 71.6°.
Vector (Cross) Product
The cross product only works in three dimensions and gives another vector that is perpendicular to both original vectors. Its length equals the area of the parallelogram formed by the two vectors.
Formula using a determinant (a fancy 3×3 table):
u × v = | i j k | | u_x u_y u_z | | v_x v_y v_z |
where i, j, k are unit vectors along the x, y, z axes. The result points out of the plane, like a flagpole standing upright from a sheet of paper.
Worked Example: Cross Product
Let p = (1, 0, 0) (pointing along x) and q = (0, 1, 0) (pointing along y).
Using the determinant, p × q = (0·0‑0·1, 0·0‑1·0, 1·1‑0·0) = (0, 0, 1). The result points along the z‑axis, straight out of the page.
Key Formulas at a Glance
| Operation | Formula |
|---|---|
| Magnitude (2‑D) | |v| = √(v_x² + v_y²) |
| Magnitude (3‑D) | |v| = √(v_x² + v_y² + v_z²) |
| Unit Vector | û = v / |v| |
| Addition | u + v = (u_x+v_x, u_y+v_y, u_z+v_z) |
| Subtraction | u – v = (u_x‑v_x, u_y‑v_y, u_z‑v_z) |
| Dot Product | u·v = u_xv_x + u_yv_y + u_zv_z = |u||v|cosθ |
| Cross Product | u×v = (u_yv_z‑u_zv_y, u_zv_x‑u_xv_z, u_xv_y‑u_yv_x) |
Quick Bullet Summary
- Vector = arrow with magnitude + direction.
- Add/subtract by component‑wise arithmetic.
- Dot product = measure of parallelism; zero means perpendicular.
- Cross product = gives a perpendicular vector; only in 3‑D.
- Unit vector shows direction alone.
📝 Likely Exam Questions
- Find the magnitude of the vector a = (5,‑12).
Answer: |a| = √(5² + (‑12)²) = √(25+144) = √169 = 13. - Given u = (2,3,1) and v = (‑1,4,2), compute u·v and the angle between them.
Answer: u·v = 2·(‑1)+3·4+1·2 = ‑2+12+2 = 12. |u|=√(4+9+1)=√14, |v|=√(1+16+4)=√21. cosθ = 12/(√14·√21). θ ≈ 38.7°. - Determine the unit vector in the direction of w = (‑3, 4).
Answer: |w| = √(9+16)=√25=5. û = (‑3/5, 4/5). - If p = (1,2,3) and q = (4,‑1,0), find p × q.
Answer: p × q = (2·0‑3·(‑1), 3·4‑1·0, 1·(‑1)‑2·4) = (3, 12, ‑9). - Show that the vectors a = (2,‑1) and b = (‑4,2) are parallel.
Answer: b = –2·a, so they have the same (or opposite) direction; thus they are parallel.