← Vectors
Mathematics · Topic Cheatsheet

Vectors

19 key results accumulated across 2 chapters.

Magnitude
Ch 1
∣v∣=v12+v22+v32|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}
Addition / scaling
Ch 1
Component-wise: (a1+b1, … )(a_1+b_1,\,\dots); kv=(kv1,… )k\mathbf{v}=(kv_1,\dots).
Dot product
Ch 1
a⋅b=a1b1+a2b2+a3b3=∣a∣∣b∣cos⁡θ\mathbf{a}\cdot\mathbf{b} = a_1b_1+a_2b_2+a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta
Perpendicular test
Ch 1
a⋅b=0  ⟺  a⊥b\mathbf{a}\cdot\mathbf{b} = 0 \iff \mathbf{a}\perp\mathbf{b}
Unit vector
Ch 1
v^=v∣v∣\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}
Magnitude / unit
Ch 1
∣a∣=a12+a22+a32,  a^=a∣a∣|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2},\;\hat{\mathbf a}=\tfrac{\mathbf a}{|\mathbf a|}
Dot product
Ch 1
a⋅b=∣a∣∣b∣cos⁡θ=a1b1+a2b2+a3b3\mathbf a\cdot\mathbf b = |\mathbf a||\mathbf b|\cos\theta = a_1b_1+a_2b_2+a_3b_3
=0 ⟺ perpendicular.
Cross product
Ch 1
∣a×b∣=∣a∣∣b∣sin⁡θ|\mathbf a\times\mathbf b| = |\mathbf a||\mathbf b|\sin\theta
⟂ to both; =0 ⟺ parallel.
Common trap
Ch 1
Dot gives a scalar, cross gives a vector; angle uses the dot product.
Vector equation of a line
Ch 2
r=a+λd\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}
a\mathbf{a} = point on line, d\mathbf{d} = direction, λ∈R\lambda\in\mathbb{R}.
Parallel lines
Ch 2
Direction vectors are scalar multiples.
Point on a line
Ch 2
Substitute and solve for a single consistent λ\lambda.
Equation of a plane
Ch 2
n⋅r=d  ⇔  n1x+n2y+n3z=d\mathbf{n}\cdot\mathbf{r} = d \;\Leftrightarrow\; n_1x + n_2y + n_3z = d
Normal vector
Ch 2
The coefficients of x,y,zx,y,z ARE the normal n\mathbf{n}; it is ⟂ to the plane.
Finding d
Ch 2
d=n⋅ad = \mathbf{n}\cdot\mathbf{a}
Using any known point a\mathbf{a} on the plane.
Line (vector form)
Ch 2
r=a+td\mathbf r = \mathbf a + t\mathbf d
a = point on line, d = direction.
Plane
Ch 2
r⋅n=a⋅n\mathbf r\cdot\mathbf n = \mathbf a\cdot\mathbf n
n = normal vector.
Intersections
Ch 2
Lines: solve for t,s (parallel if directions ∥; skew if no solution in 3D).
Common trap
Ch 2
Two lines can be skew (never meet, not parallel) in 3D — no 2D analogue.