O Level Elementary Mathematics: Vectors Worked Examples

Quick answer

Vectors in O Level Mathematics often trip students up because they look different from the examples you’ve practised. But don’t worry. Once you understand how to break down the problem step by step, you’ll find that most of the methods you need are familiar. This guide will help you tackle vectors questions without freezing up.

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What you need to know

A vector is a mathematical object with both magnitude (size) and direction. In simpler terms, it tells you not only how far something moves but also where it's moving to. You'll usually see them represented as arrows or in coordinates like (𝑥, 𝑦). In O Level exams, vectors often appear in questions that require you to find resultant vectors, magnitudes, or directions.

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Key concepts

Recognising vector notation

In exams, vectors are often presented in bold (like u) or with an arrow on top (like $\overrightarrow{AB}$). They can also be written in coordinate form, such as (2, 3), meaning 2 units in the x-direction and 3 units in the y-direction.

Vector addition and subtraction

To add or subtract vectors, you simply add or subtract their corresponding components. For example, if a = (2, 3) and b = (1, 4), then a + b = (2+1, 3+4) = (3, 7).

Multiplying a vector by a scalar

When you multiply a vector by a scalar (a single number), you multiply each component of the vector by that number. For instance, if you have 2a and a = (2, 3), then 2a = (22, 23) = (4, 6).

Common mistakes students make

1. Freezing when the question looks different: Remember, vectors are just like directions on a map. Break them down into steps you know.
2. Rushing algebra steps: This is where many students lose unnecessary marks. Slow down and check each component.
3. Ignoring vector direction: Sometimes students calculate the magnitude but forget the direction. Always check both parts.

Exam tip

When you see a vector question, quickly sketch a diagram if possible. It helps to visualise the problem and often reveals the solution. Remember, clear working and neat presentation can gain you marks even if the final answer is wrong.

Worked examples

Question 1

Find the resultant vector of u = (4, -3) and v = (-1, 5).

Solution

Step 1: Add the corresponding components of the vectors.
Why: This gives you the resultant vector, which is like finding the total movement in both directions.

u + v = (4 + (-1), -3 + 5)

Step 2: Simplify the expression.
Why: Simplification gives you the final answer in a clean, readable form.

Resultant vector = (3, 2)

Question 2

Calculate the magnitude of the vector w = (3, 4).

Solution

Step 1: Use the formula for magnitude, $\sqrt{𝑥^2 + 𝑦^2}$.
Why: This formula gives you the length of the vector, similar to finding the length of the hypotenuse in a right triangle.

Magnitude of w = $\sqrt{3^2 + 4^2}$

Step 2: Calculate the square of each component and add them.
Why: This step is necessary to apply the Pythagorean theorem.

= $\sqrt{9 + 16}$

Step 3: Take the square root of the sum.
Why: The square root gives you the actual length of the vector.

= $\sqrt{25} = 5$

Question 3

Determine the vector p if 3p = (6, 9).

Solution

Step 1: Divide each component by the scalar 3.
Why: This reverses the multiplication by a scalar to find the original vector.

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p = $\left(\frac{6}{3}, \frac{9}{3}\right)$

Step 2: Simplify the components.
Why: Simplification gives you the final answer in a clean, readable form.

p = (2, 3)

Question 4

If a = (2, 3) and b = (5, -2), find the vector from a to b.

Solution

Step 1: Subtract the coordinates of a from b.
Why: This gives you the vector pointing from one point to another.

Vector from a to b = (5 - 2, -2 - 3)

Step 2: Simplify the expression.
Why: Simplification gives you the final answer in a clean, readable form.

= (3, -5)

Quick summary

• Vectors have both direction and magnitude.
• Add/subtract vectors by their components.
• Multiply vectors by scalars by multiplying each component.
• Use the formula $\sqrt{𝑥^2 + 𝑦^2}$ to find vector magnitude.
• Draw diagrams to visualise vector problems.

FAQ

Q 1: What is a vector?
A vector is a quantity that has both magnitude (how much) and direction (where to).

Q 2: How do I find the magnitude of a vector?
Use the formula $\sqrt{𝑥^2 + 𝑦^2}$, which is like finding the length of the hypotenuse in a triangle.

Q 3: What does it mean to multiply a vector by a scalar?
It means you multiply each component of the vector by the scalar number, changing the vector's magnitude but not its direction.

Q 4: How can I visualise vectors better?
Draw them as arrows on a graph. The length shows magnitude, and the arrow points in the direction.

Related Topics You Should Learn Next

• O Level Elementary Mathematics: Vectors and How to Master Them
• O Level Elementary Mathematics: Vectors Explained Simply
• O Level EMath Vectors Guide
• Learn More About Vectors

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