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Vectors

Chapter 1 · Vector Foundations

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Magnitude
∣v∣=v12+v22+v32|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}
Addition / scaling
Component-wise: (a1+b1, … )(a_1+b_1,\,\dots); kv=(kv1,… )k\mathbf{v}=(kv_1,\dots).
Dot product
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
a⋅b=0  ⟺  a⊥b\mathbf{a}\cdot\mathbf{b} = 0 \iff \mathbf{a}\perp\mathbf{b}
Unit vector
v^=v∣v∣\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}
Magnitude / unit
∣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
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
∣a×b∣=∣a∣∣b∣sin⁡θ|\mathbf a\times\mathbf b| = |\mathbf a||\mathbf b|\sin\theta
⟂ to both; =0 ⟺ parallel.
Common trap
Dot gives a scalar, cross gives a vector; angle uses the dot product.