Exercise 3.1 — Q1: Vectors in Space

Rectangular Coordinate System in Space

In 3D space, every point $P$ is located by an ordered triple $(x,y,z)$ with respect to three mutually perpendicular axes:

• $x$-axis — horizontal (left–right)
• $y$-axis — horizontal (front–back)
• $z$-axis — vertical (up–down)

The origin $O=(0,0,0)$ is the intersection of all three axes.

Unit Vectors and Components

Three standard unit vectors define the positive directions of the axes: $\hat{i}=(1,0,0),\hat{j}=(0,1,0),\hat{k}=(0,0,1)$

Any vector $\vec{v}$ in space can be written in component form: $\vec{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$ where $v_1$, $v_2$, $v_3$ are the scalar components along the $x$-, $y$-, and $z$-axes respectively.

Magnitude of a Vector

The magnitude (length) of $\vec{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$ is: $|\vec{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$

Vector Between Two Points

Given points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$, the vector from $P$ to $Q$ is: $\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$

Parallel Vectors

Vectors $\mathbf{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\mathbf{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ are parallel if: $\mathbf{a}=k\mathbf{b}⟺\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$ for some scalar $k\ne 0$. If $k>0$, the vectors point in the same direction; if $k<0$, they are antiparallel.

Properties of Vector Addition

For vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ in space:

The null vector $\vec{0}=0\hat{i}+0\hat{j}+0\hat{k}$ has zero magnitude and acts as the additive identity.