Welcome to Problem Solving with Vectors!
In this chapter, we are going to take everything you’ve learned about what vectors are and start using them to solve real-world puzzles. Think of vectors as the ultimate GPS instructions—they don’t just tell you where something is, but exactly how to get there, how fast to move, and what forces are pushing against you along the way.
Don’t worry if this seems tricky at first! We are going to break it down step-by-step. By the end of these notes, you’ll see that vector problem solving is mostly about drawing a good picture and following a logical path.
1. Vectors in Pure Mathematics: Geometric Proofs
One of the most common ways to use vectors in "Pure" math is to prove things about shapes. You can use vectors to show that two lines are parallel or to find where two lines meet.
The Secret of "Path-Finding"
To find a vector between two points, you can take any path you like! If you want to go from point \(A\) to point \(B\), but you only know the vectors to the origin (\(O\)), you can go "the long way around":
\(\vec{AB} = \vec{AO} + \vec{OB}\)
Since \(\vec{AO}\) is just the negative of \(\vec{OA}\), we often write this as:
\(\vec{AB} = \mathbf{b} - \mathbf{a}\)
Checking for Parallel Lines
How do you know if two vectors are parallel? It’s simple: one must be a "multiple" of the other. If you have vector \(\mathbf{u}\) and vector \(\mathbf{v}\), they are parallel if:
\(\mathbf{u} = k\mathbf{v}\)
Example: The vector \(\begin{pmatrix} 2 \\ 4 \end{pmatrix}\) is parallel to \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) because it is exactly double the size, just pointing in the same direction.
Quick Review: Geometric Rules
- Midpoints: If \(M\) is the midpoint of line \(AB\), then \(\vec{AM} = \frac{1}{2}\vec{AB}\).
- Collinear Points: If points \(A, B,\) and \(C\) lie on a straight line, the vectors \(\vec{AB}\) and \(\vec{BC}\) must be parallel.
Key Takeaway: In geometry problems, always express your "unknown" path in terms of vectors you already know. If one vector is a scalar multiple of another, they are parallel!
2. Vectors in Context: Forces and Equilibrium
In the real world, vectors represent Forces. Imagine two people pulling a heavy crate in different directions. The resultant force is simply the vector sum of those two pulls.
Resultant Forces
To find the total force acting on an object, you just add the individual vectors together:
\(\mathbf{R} = \mathbf{F_1} + \mathbf{F_2} + \dots + \mathbf{F_n}\)
The Concept of Equilibrium
If an object is stationary (not moving) or moving at a constant velocity, the forces are in equilibrium. This is a fancy way of saying they all cancel each other out.
Key Point: In equilibrium, the sum of all force vectors is zero.
\(\sum \mathbf{F} = \mathbf{0}\)
Example: If a boat is being pulled by two tugboats with forces \(\mathbf{F_1} = \begin{pmatrix} 50 \\ 20 \end{pmatrix}\) and \(\mathbf{F_2} = \begin{pmatrix} -10 \\ 30 \end{pmatrix}\), the resultant force is \(\begin{pmatrix} 40 \\ 50 \end{pmatrix}\).
Did you know?
Civil engineers use these exact vector calculations to make sure bridges don't fall down. They ensure that all the forces (gravity, wind, weight of cars) sum to zero so the bridge stays in equilibrium!
Key Takeaway: Adding force vectors gives you the "resultant" force. If the object isn't accelerating, the vectors must add up to \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\).
3. Vectors in Kinematics: Motion in 2D and 3D
Vectors are perfect for describing how objects move. Instead of just saying "the car is going 30 mph," we can say exactly which direction it's heading.
Position vs. Displacement
- Position Vector (\(\mathbf{r}\)): Where an object is relative to a fixed origin (usually \(O\)).
- Displacement (\(\mathbf{s}\)): How far and in what direction the object has moved from its starting point.
The relationship is:
\(\mathbf{r} = \mathbf{r_0} + \mathbf{s}\)
(Current Position = Starting Position + Displacement)
Constant Velocity
If an object moves with a constant velocity (\(\mathbf{v}\)), its displacement after time \(t\) is simply \(\mathbf{v} \times t\).
The position formula becomes:
\(\mathbf{r} = \mathbf{r_0} + \mathbf{v}t\)
Vector SUVAT (Constant Acceleration)
When an object is accelerating, we use the standard equations of motion, but in bold vector form! The most common ones you'll use are:
1. \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\)
2. \(\mathbf{s} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\)
3. \(\mathbf{r} = \mathbf{r_0} + \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\)
Common Mistake to Avoid: You cannot use the formula \(v^2 = u^2 + 2as\) with vectors directly because you can't "square" a vector in the same way you square a number. Stick to the formulas above!
Key Takeaway: To find where an object is at time \(t\), find its displacement (\(\mathbf{s}\)) and add it to its starting position (\(\mathbf{r_0}\)).
4. Working with Components: \(\mathbf{i}, \mathbf{j}, \mathbf{k}\)
While column vectors \(\begin{pmatrix} x \\ y \end{pmatrix}\) are great for calculations, the syllabus also requires you to be comfortable with unit vector notation.
- \(\mathbf{i}\) is one unit in the \(x\)-direction (horizontal).
- \(\mathbf{j}\) is one unit in the \(y\)-direction (vertical).
- \(\mathbf{k}\) is one unit in the \(z\)-direction (depth, for 3D).
Analogy: Think of \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) as "steps." A vector \(3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}\) just means "Take 3 steps right, 2 steps down, and 5 steps forward."
Quick Review: Magnitude and Distance
To find the magnitude (size) of a vector, use Pythagoras:
For \(\mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\), the magnitude is \(|\mathbf{a}| = \sqrt{x^2 + y^2 + z^2}\).
The distance between two position vectors \(\mathbf{a}\) and \(\mathbf{b}\) is the magnitude of the vector connecting them: \(|\mathbf{b} - \mathbf{a}|\).
Key Takeaway: \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are just labels for directions. When adding them, only add the \(\mathbf{i}\)s to \(\mathbf{i}\)s, \(\mathbf{j}\)s to \(\mathbf{j}\)s, and \(\mathbf{k}\)s to \(\mathbf{k}\)s.
5. Problem Solving Strategy: The "Golden Rules"
When you face a tough exam question, follow this checklist:
- Sketch it: Even a rough drawing of the vectors can stop you from making a direction error (like forgetting a minus sign).
- Label your Origin: Decide where \((0,0)\) is. Usually, it's the starting point of the object.
- Identify the variables: Write down what you know (\(\mathbf{u}, \mathbf{a}, t, \mathbf{r_0}\)).
- Choose your formula: Pick a vector SUVAT equation or a geometric path.
- Solve component by component: If you have an equation like \(\begin{pmatrix} x \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ y \end{pmatrix} + t\begin{pmatrix} 1 \\ -1 \end{pmatrix}\), split it into two simple equations (one for \(x\), one for \(y\)).
Key Takeaway: Vectors allow you to solve two (or three) problems at once! Treat the horizontal and vertical components as separate equations that are linked by time (\(t\)).
Summary Table
Concept: Parallel Vectors
Math: \(\mathbf{a} = k\mathbf{b}\)
Concept: Equilibrium
Math: \(\mathbf{F_1} + \mathbf{F_2} + \dots = \mathbf{0}\)
Concept: Position at time \(t\)
Math: \(\mathbf{r} = \mathbf{r_0} + \mathbf{v}t\) (Constant \(v\))
Concept: Magnitude (Speed/Distance)
Math: \(\sqrt{x^2 + y^2 + z^2}\)
Keep practicing! Vectors can feel like learning a new language, but once you speak "Vector," you'll find it's one of the most powerful tools in your mathematical toolkit. You've got this!