Vectors

Introduction

A vector is a quantity with both magnitude and direction, such as displacement, velocity or force; a plain number with magnitude only is a scalar. In coordinates we write a vector in space as $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, where $\hat{i}, \hat{j}, \hat{k}$ are the unit vectors along the axes. Its magnitude is $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$. This chapter studies when vectors point along one line or lie in one plane, how any vector can be built from a few basic ones, and the scalar (dot) product together with its geometry and uses.

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Collinear and coplanar vectors

Collinear vectors

Two vectors are collinear (parallel) if they lie along the same line or parallel lines, that is, if one is a scalar multiple of the other:

$$\vec{a} \parallel \vec{b} \iff \vec{b} = \lambda,\vec{a} \ \text{ for some scalar } \lambda.$$

In components, $\vec{a} = a_1\hat i + a_2\hat j + a_3\hat k$ and $\vec{b} = b_1\hat i + b_2\hat j + b_3\hat k$ are collinear when their components are proportional, $\dfrac{b_1}{a_1} = \dfrac{b_2}{a_2} = \dfrac{b_3}{a_3}$. Three points $A, B, C$ are collinear when $\vec{AB} = \lambda,\vec{AC}$ for some scalar $\lambda$.

Coplanar vectors

Vectors are coplanar if they lie in (or are parallel to) one common plane. Any two vectors are always coplanar; the real question is about three. Three vectors $\vec a, \vec b, \vec c$ are coplanar exactly when one of them can be written as a linear combination of the other two, or equivalently when their scalar triple product vanishes:

$$\vec a \cdot (\vec b \times \vec c) = \begin{vmatrix} a_1 & a_2 & a_3\ b_1 & b_2 & b_3\ c_1 & c_2 & c_3 \end{vmatrix} = 0.$$

Note that all collinear vectors are automatically coplanar, but coplanar vectors need not be collinear.

Worked example. Show that $\vec a = \hat i + 2\hat j$, $\vec b = \hat j + \hat k$ and $\vec c = \hat i + 3\hat j + \hat k$ are coplanar. The triple product is

$$ \begin{aligned} \begin{vmatrix} 1 & 2 & 0\ 0 & 1 & 1\ 1 & 3 & 1 \end{vmatrix} &= 1(1\cdot 1 - 1\cdot 3) - 2(0\cdot 1 - 1\cdot 1) + 0 \ &= 1(-2) - 2(-1) + 0 = -2 + 2 = 0, \end{aligned} $$

so the three vectors are coplanar. Indeed $\vec c = \vec a + \vec b$, a linear combination of the other two.

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