Welcome to Vectors in Two Dimensions!
Hello! This chapter introduces you to the fascinating world of Vectors. If you sometimes find coordinate geometry a bit dry, vectors offer a powerful, visual, and often simpler way to deal with movement, direction, and geometry.
Don't worry if this seems tricky at first. Vectors are used everywhere—from physics (describing forces and velocities) to computer graphics. By the end of these notes, you’ll be able to calculate magnitudes, navigate directions, and solve complex geometric problems using straightforward algebra!
1. The Fundamentals: Scalars vs. Vectors
What is a Vector?
A vector is a quantity that has both magnitude (size/length) and direction.
- Scalar: Only has magnitude (e.g., time, mass, temperature, speed).
- Vector: Has magnitude and direction (e.g., displacement, force, velocity).
Vector Notation (How we write them)
In Additional Maths, you need to be familiar with several ways of writing a vector, which represents a movement in the x and y directions.
(a) Bold Lowercase Letters (\(\mathbf{a}\) or \(\mathbf{p}\))
When you see a variable in bold, like \(\mathbf{a}\) or \(\mathbf{p}\), it represents a vector. If you are writing by hand, you should underline it, e.g., a, since you cannot write in bold.
(b) Arrow Notation (\(\vec{AB}\))
This notation describes the vector that starts at point A and ends at point B. The direction matters!
- \(\vec{AB}\) means A to B.
- \(\vec{BA}\) means B to A. Notice that \(\vec{BA} = - \vec{AB}\).
(c) Column Vector Form (\(\begin{pmatrix} x \\ y \end{pmatrix}\))
This is the most common form for calculations in two dimensions:
Analogy: Think of this as driving directions. The top number (\(x\)) is the movement horizontally (East/West), and the bottom number (\(y\)) is the movement vertically (North/South).
(d) Unit Vector Form (\(\mathbf{i}, \mathbf{j}\))
The vectors \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors (vectors of length 1) used to describe the primary directions:
- \(\mathbf{i}\) is the unit vector in the positive x-direction (\(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\)).
- \(\mathbf{j}\) is the unit vector in the positive y-direction (\(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\)).
Therefore, the vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\) can also be written as:
Quick Review: Correct notation is vital. Always use bold or underlined letters for vectors, and remember that \(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\) is the same as \(5\mathbf{i} - 2\mathbf{j}\).
2. Position and Displacement Vectors
Position Vectors (13.2)
A position vector describes the location of a specific point in space relative to the Origin (O).
- The position vector of point A is \(\vec{OA}\), usually written as \(\mathbf{a}\).
- If point A has coordinates (3, 5), its position vector is \(\mathbf{a} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}\).
Displacement Vector Between Two Points
To find the vector representing the movement from point A to point B, you subtract the position vector of the starting point (A) from the position vector of the end point (B).
Memory Aid: The vector \(\vec{AB}\) is always "B minus A" (End minus Start).
Example: If \(\mathbf{a} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 6 \\ 7 \end{pmatrix}\), then:
\(\vec{AB} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} 6 \\ 7 \end{pmatrix} - \begin{pmatrix} 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 6-4 \\ 7-1 \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}\)
Key Takeaway: Position vectors start at the origin. Displacement vectors (like \(\vec{AB}\)) describe the journey between two points.
3. Calculating with Vectors
(a) Magnitude of a Vector (Length) (13.3)
The magnitude of a vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), denoted by \(|\mathbf{a}|\), is its length. We use the Pythagorean Theorem (since x and y components form a right-angled triangle).
Did you know? Finding the magnitude of \(\vec{AB}\) is exactly the same as using the distance formula between points A and B in coordinate geometry!
Example: Find the magnitude of \(\mathbf{v} = 3\mathbf{i} - 4\mathbf{j}\).
\(|\mathbf{v}| = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) units.
(b) Unit Vectors (13.2)
A unit vector is a vector that has a magnitude of exactly 1.
To find the unit vector in the same direction as \(\mathbf{a}\) (we call this \(\hat{\mathbf{a}}\)), you take the original vector and divide it by its magnitude. This effectively "scales" the vector down to length 1.
Example: Find the unit vector for \(\mathbf{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\) (where \(|\mathbf{v}| = 5\)):
\(\hat{\mathbf{v}} = \frac{1}{5} \begin{pmatrix} 3 \\ -4 \end{pmatrix} = \begin{pmatrix} 3/5 \\ -4/5 \end{pmatrix}\)
(c) Addition and Subtraction (13.3)
Adding and subtracting vectors is simple: you just add or subtract the corresponding components.
- Addition: \(\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}\)
- Subtraction: \(\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} - \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}\)
Geometrically: Vector addition follows the "head-to-tail" rule. If you go along \(\mathbf{a}\) and then along \(\mathbf{b}\), the resultant vector is \(\mathbf{a} + \mathbf{b}\).
(d) Multiplication by a Scalar (13.3)
Multiplying a vector by a scalar (a regular number, \(k\)) changes its magnitude (length) but keeps the direction the same (unless \(k\) is negative).
Key Concept: Parallel Vectors
If two vectors, \(\mathbf{a}\) and \(\mathbf{b}\), are parallel, one is simply a scalar multiple of the other:
(e) Equating Like Vectors (13.3)
If two vectors are equal, their corresponding components must be equal. This is often used to solve for unknown variables.
If \(\begin{pmatrix} 2p \\ 3+q \end{pmatrix} = \begin{pmatrix} 10 \\ 7 \end{pmatrix}\):
- x-components: \(2p = 10 \implies p = 5\)
- y-components: \(3+q = 7 \implies q = 4\)
Common Mistake to Avoid: When finding the magnitude, do not forget the square root at the end! \(|\mathbf{a}|\) is a scalar (a number), not a vector.
4. Solving Geometric Problems with Vectors
Vectors provide a robust way to prove geometric properties like collinearity or parallel lines.
(a) Collinearity (Points on a Straight Line)
Three points, A, B, and C, are collinear if they lie on the same straight line.
To prove A, B, and C are collinear, you must show two things:
- \(\vec{AB}\) is parallel to \(\vec{BC}\). (i.e., \(\vec{AB} = k\vec{BC}\) or \(\vec{AB} = k\vec{AC}\))
- A common point (like B or A) exists between the two vectors.
If they are parallel and share a point, they must lie on the same line!
(b) Using Ratios in Vector Geometry
If P divides the line segment AB in the ratio 1:2, this means P is closer to A, and \(\vec{AP}\) is one-third of the total distance \(\vec{AB}\).
To find the position vector of P, \(\mathbf{p}\):
\(\mathbf{p} = \vec{OA} + \vec{AP} = \mathbf{a} + \frac{1}{3} (\mathbf{b} - \mathbf{a})\)
Strategy for complex problems: If a diagram is provided (or if you can draw one), always try to express the unknown vector by traveling through known vectors. For example, to find \(\vec{XY}\), try the route \(\vec{XO} + \vec{OY}\).
Key Takeaway: Vector geometry relies on the fact that if \(\mathbf{p} = k\mathbf{q}\), the vectors are parallel. If they share a common point, the points are collinear.
5. Vectors in Motion: Velocity and Kinematics
In physics and Additional Mathematics, movement is perfectly described by vectors, especially when considering speed and direction simultaneously. This is often called kinematics.
(a) Velocity, Displacement, and Speed
- Displacement (\(\mathbf{s}\)): The position vector of the object at time \(t\).
- Velocity (\(\mathbf{v}\)): The rate of change of displacement. This is a vector because it includes the direction of motion.
-
Speed: This is a scalar quantity—it is simply the magnitude of the velocity:
Speed = \(|\mathbf{v}|\)
(b) Composition and Resultant Vectors (13.4)
When an object is affected by multiple motions (like a boat in a current or a plane in the wind), we use vector addition to find the overall motion, known as the resultant vector.
Example: A boat tries to move across a river (velocity \(\mathbf{v}_B\)), but the river has a current (velocity \(\mathbf{v}_C\)).
The resultant velocity, \(\mathbf{v}_R\), is the vector sum:
If \(\mathbf{v}_B = \begin{pmatrix} 5 \\ 0 \end{pmatrix}\) and \(\mathbf{v}_C = \begin{pmatrix} 0 \\ 2 \end{pmatrix}\), then the resultant velocity is \(\mathbf{v}_R = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\). The actual speed of the boat would be \(|\mathbf{v}_R| = \sqrt{5^2 + 2^2} = \sqrt{29}\).
(c) Solving Collision Problems (Contextual Problems)
A common application involves two particles, P and Q, moving according to their velocity vectors.
If a particle starts at position \(\mathbf{s}_0\) and moves with constant velocity \(\mathbf{v}\), its displacement \(\mathbf{s}_t\) at time \(t\) is given by:
Condition for Collision: If two particles, P and Q, collide, they must be at the same physical location at the same time \(t\).
To solve this:
- Write out the position vector equation for P in terms of \(t\).
- Write out the position vector equation for Q in terms of \(t\).
- Set the two equations equal to each other.
- Equate the corresponding \(\mathbf{i}\) (x) components to find \(t\).
- Equate the corresponding \(\mathbf{j}\) (y) components to verify the time \(t\).
Quick Tip for Struggling Students: When working through word problems, immediately translate all information into column vectors. A velocity of "3 m/s East" is \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). A starting position at (1, 4) means \(\mathbf{s}_0 = \begin{pmatrix} 1 \\ 4 \end{pmatrix}\).
Key Takeaway: Velocity problems are just vector addition problems solved in a context. Use the formula \(\mathbf{s}_t = \mathbf{s}_0 + t\mathbf{v}\) and remember that speed is the magnitude of the velocity vector.
Chapter Summary: Key Formulas to Memorise
- Displacement \(\vec{AB}\): \(\mathbf{b} - \mathbf{a}\)
- Magnitude \(|\mathbf{a}|\): \(\sqrt{x^2 + y^2}\)
- Unit Vector \(\hat{\mathbf{a}}\): \(\frac{\mathbf{a}}{|\mathbf{a}|}\)
- Position at time \(t\): \(\mathbf{s}_t = \mathbf{s}_0 + t\mathbf{v}\)
- Parallel Condition: \(\mathbf{a} = k\mathbf{b}\)