3.1 Vector Addition and Components

Rectangular Coordinate System in Space

In three-dimensional space, three mutually perpendicular axes — the $x$-axis, $y$-axis, and $z$-axis — intersect at a point called the origin $O$. Any point $P$ in space is uniquely identified by an ordered triple $(x,y,z)$:

• $x$ = perpendicular distance from the $yz$-plane
• $y$ = perpendicular distance from the $xz$-plane
• $z$ = perpendicular distance from the $xy$-plane

This extends the familiar 2D coordinate system (the $xy$-plane) into 3D space.

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Unit Vectors and Components

Three standard unit vectors are defined along the coordinate axes:

Any vector $\mathbf{v}$ in space can be written in component form: $\mathbf{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$ where $v_1$, $v_2$, $v_3$ are the components (scalar projections) along each axis.

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Magnitude of a Vector

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

The unit vector in the direction of $\mathbf{v}$ is: $\hat{v}=\frac{\mathbf{v}}{|\mathbf{v}|}$

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Vector Operations in Space

Position Vector and Vector Between Two Points

The position vector of point $P(x,y,z)$ from the origin is $\overrightarrow{OP}=x\hat{i}+y\hat{j}+z\hat{k}$.

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

Scalar Multiplication

For scalar $k$ and vector $\mathbf{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$: $k\mathbf{a}=k{a}_1\hat{i}+k{a}_2\hat{j}+k{a}_3\hat{k}$

A vector of magnitude $m$ in the opposite direction of $\mathbf{a}$ is: $-m\hat{a}=-m\frac{\mathbf{a}}{|\mathbf{a}|}$

Parallel Vectors

Vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel if $\mathbf{a}=k\mathbf{b}$ for some scalar $k$, equivalently: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$

Collinear Points

Three points $A$, $B$, $C$ are collinear if $\overrightarrow{AB}=k\overrightarrow{BC}$ for some scalar $k$ (the vectors are parallel and share a common point).

Parallelogram Vertices

In parallelogram $ABCD$, opposite sides are equal and parallel: $\overrightarrow{AB}=\overrightarrow{DC},\overrightarrow{AD}=\overrightarrow{BC}$ This is used to find a missing fourth vertex given three vertices.

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Properties of Vector Addition

Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ be any vectors in space.

1. Commutative Law

$\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}$ Proof: In component form, $(a_1+b_1)\hat{i}+\cdots =(b_1+a_1)\hat{i}+\cdots$ since addition of real numbers is commutative.

2. Associative Law

$(\mathbf{a}+\mathbf{b})+\mathbf{c}=\mathbf{a}+(\mathbf{b}+\mathbf{c})$ Proof: Follows from the associativity of real number addition applied component-wise.

3. Identity Element (Null Vector)

The null vector $0=0\hat{i}+0\hat{j}+0\hat{k}$ satisfies: $\mathbf{a}+0=0+\mathbf{a}=\mathbf{a}$

4. Additive Inverse

For every vector $\mathbf{a}$, its additive inverse is $-\mathbf{a}$: $\mathbf{a}+(-\mathbf{a})=0$

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Summary Table