Overview
This lecture covers vector notation, types of vectors, operations, expressing vectors in terms of other vectors, position vectors, unit vectors, and solving vector-related problems using formulas and laws.
Vector Notation & Representation
- Vectors are often denoted by bold letters (e.g., a, b) or underlined when handwritten.
- Vectors from one point to another are written with an arrow, e.g., (\overrightarrow{AC}).
- The magnitude (length) of a vector is shown with modulus symbols, e.g., (|\vec{a}|).
Types of Vectors
- Equal vectors have the same direction and magnitude.
- Negative vectors have the same magnitude but opposite directions.
- Zero vectors have zero magnitude.
- Scalar multiplication: if (\vec{b} = k\vec{a}), direction depends on the sign of (k); positive (k) is same direction, negative (k) is opposite.
- Unit vectors have a magnitude of 1.
Unit Vectors
- The unit vector in the direction of (\vec{a}) is (\hat{a} = \vec{a} / |\vec{a}|).
- Example: For (\vec{a} = 3\mathbf{i} + 4\mathbf{j}), magnitude is 5, so unit vector is ((3/5)\mathbf{i} + (4/5)\mathbf{j}).
Vector Addition & Subtraction
- Triangle Law: (\overrightarrow{PQ} + \overrightarrow{QR} = \overrightarrow{PR}).
- Parallelogram Law: (\overrightarrow{PQ} + \overrightarrow{PS} = \overrightarrow{PR}).
- Commutative law: (\vec{a} + \vec{b} = \vec{b} + \vec{a}).
- Subtraction: (\overrightarrow{QR} = \overrightarrow{PR} - \overrightarrow{PQ}).
Expressing Vectors & Position Vectors
- Any vector in a plane formed by non-parallel vectors (\vec{a}) and (\vec{b}) is (m\vec{a} + n\vec{b}).
- If (p\vec{a} + q\vec{b} = r\vec{a} + s\vec{b}), then (p = r) and (q = s).
- A position vector describes the location of a point relative to the origin (e.g., (\overrightarrow{OA})).
Collinear Points
- Points are collinear if they lie on the same line, implying a scalar multiple relationship between their vectors.
Vectors in Cartesian Plane & IJ Notation
- Unit vectors (\mathbf{i}) and (\mathbf{j}) point in the positive x and y directions, respectively.
- A vector to point (P(x, y)) is (x\mathbf{i} + y\mathbf{j}).
- The magnitude is (\sqrt{x^2 + y^2}).
Distance Between Two Points
- Distance between (P(x_1, y_1)) and (Q(x_2, y_2)) is (\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).
Example Calculations
- To find (\overrightarrow{PQ}), subtract position vectors: (Q - P).
- Magnitude of (3\mathbf{i} + 4\mathbf{j}) is 5; unit vector is ((3/5)\mathbf{i} + (4/5)\mathbf{j}).
- For a vector of magnitude 65 in the same direction, multiply the unit vector by 65.
Key Terms & Definitions
- Vector — A quantity with both magnitude and direction.
- Magnitude — The length of a vector.
- Unit Vector — A vector with magnitude 1 indicating direction.
- Position Vector — A vector from the origin to a point.
- Collinear Points — Points that lie on the same straight line.
Action Items / Next Steps
- Review the triangular and parallelogram laws of vector addition.
- Practice expressing vectors in ij notation and calculating magnitudes.
- Attempt the practice questions on vector operations from your textbook.