Vectors - Formulas & Operations
IB Mathematics Analysis & Approaches (SL & HL)
📐 Vector Notation & Representation
Column Vector (2D):
\[\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}\]
Column Vector (3D):
\[\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\]
Component Form:
\[\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}\]
where \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are unit vectors in x, y, z directions
📍 Position & Displacement Vectors
Position Vector:
\[\overrightarrow{OA} = \mathbf{a} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\]
Vector from origin O to point A
Displacement Vector:
\[\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\]
Vector from point A to point B
➕ Vector Operations
Vector Addition:
\[\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix}\]
Vector Subtraction:
\[\mathbf{u} - \mathbf{v} = \begin{pmatrix} u_1 - v_1 \\ u_2 - v_2 \\ u_3 - v_3 \end{pmatrix}\]
Scalar Multiplication:
\[k\mathbf{v} = \begin{pmatrix} kv_1 \\ kv_2 \\ kv_3 \end{pmatrix}\]
where \(k\) is a scalar (real number)
📏 Magnitude (Length) of a Vector
2D Vector Magnitude:
\[|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}\]
3D Vector Magnitude:
\[|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\]
Given in formula booklet
Distance Between Two Points:
\[|\overrightarrow{AB}| = |\mathbf{b} - \mathbf{a}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]
🎯 Unit Vectors
Definition:
A unit vector has magnitude equal to 1
Unit Vector in Direction of \(\mathbf{v}\):
\[\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}\]
Standard Unit Vectors:
\[\mathbf{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\]
⚡ Scalar (Dot) Product
Component Form:
\[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\]
Given in formula booklet
Geometric Form:
\[\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta\]
where \(\theta\) is the angle between the vectors
Given in formula booklet
Important Properties:
• If \(\mathbf{u} \cdot \mathbf{v} = 0\), vectors are perpendicular
• \(\mathbf{v} \cdot \mathbf{v} = |\mathbf{v}|^2\)
• Commutative: \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\)
📐 Angle Between Two Vectors
Formula:
\[\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\]
Given in formula booklet
To Find the Angle:
\[\theta = \arccos\left(\frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\right)\]
📍 Vector Equation of a Line
Vector Form:
\[\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\]
where \(\mathbf{a}\) = position vector of a point on the line,
\(\mathbf{b}\) = direction vector, \(\lambda\) = parameter
Given in formula booklet
Parametric Form:
\[x = x_0 + \lambda l, \quad y = y_0 + \lambda m, \quad z = z_0 + \lambda n\]
Given in formula booklet
Cartesian Form:
\[\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}\]
⚖️ Parallel & Perpendicular Vectors
Parallel Vectors:
\[\mathbf{u} = k\mathbf{v}\]
where \(k\) is a scalar. One is a scalar multiple of the other.
Perpendicular (Orthogonal) Vectors:
\[\mathbf{u} \cdot \mathbf{v} = 0\]
Scalar product equals zero
✖️ Vector (Cross) Product (HL)
Definition:
\[\mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_2v_3 - u_3v_2 \\ u_3v_1 - u_1v_3 \\ u_1v_2 - u_2v_1 \end{pmatrix}\]
Given in formula booklet
Magnitude of Cross Product:
\[|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta\]
Given in formula booklet
Properties:
• Result is perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\)
• Anti-commutative: \(\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})\)
• If \(\mathbf{u} \times \mathbf{v} = \mathbf{0}\), vectors are parallel
🛫 Vector Equation of a Plane (HL)
Vector Form:
\[\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}\]
where \(\mathbf{a}\) = position vector of a point on the plane,
\(\mathbf{b}, \mathbf{c}\) = two non-parallel direction vectors,
\(\lambda, \mu\) = parameters
Scalar (Cartesian) Form:
\[\mathbf{r} \cdot \mathbf{n} = d \quad \text{or} \quad ax + by + cz = d\]
where \(\mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\) is normal to the plane
💡 Common Applications
Area of Parallelogram:
\[\text{Area} = |\mathbf{u} \times \mathbf{v}|\]
Area of Triangle:
\[\text{Area} = \frac{1}{2}|\mathbf{u} \times \mathbf{v}|\]
Projection of \(\mathbf{u}\) onto \(\mathbf{v}\):
\[\text{proj}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}\]
💡 Exam Tip: Most vector formulas are given in the IB formula booklet, including magnitude, scalar product, vector product, and equations of lines. Always check if vectors are perpendicular (dot product = 0) or parallel (one is scalar multiple of other). Use your GDC for calculations. Remember: scalar product gives a number, vector product gives a vector. Label your diagrams clearly!