Components and resolved parts of a vector

Edexcel International GCSE Further Pure Mathematics· S7 7B· 12 min read

1. Component Form of Vectors Using i and j Base Units★★☆☆☆⏱ 4 min

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📘 Definition

Component form

A vector is written in component form as $a i + b j$ , where $a$ is the horizontal displacement and $b$ is the vertical displacement from the start to end point of the vector. $i$ is the unit vector (magnitude 1) in the positive x-direction, $j$ is the unit vector in the positive y-direction.

Example:

A vector that moves 3 units right and 2 units up is written as $3 i + 2 j$

If you are given the coordinates of the start and end points of a vector, subtract the start coordinates from the end coordinates to find the i and j components. For a vector from point $(x_{1}, y_{1})$ to $(x_{2}, y_{2})$ : the i component is $x_{2} - x_{1}$ , the j component is $y_{2} - y_{1}$ .

📐 Worked Example

Find the component form of the vector from point $P (2, 5)$ to point $Q (- 1, 7)$ .

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Step 1: Calculate the horizontal i component: subtract the x-coordinate of P from the x-coordinate of Q.
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$i\text{ component} = x_Q - x_P = -1 - 2 = -3$
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Step 2: Calculate the vertical j component: subtract the y-coordinate of P from the y-coordinate of Q.
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$j\text{ component} = y_Q - y_P = 7 - 5 = 2$
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Step 3: Combine the components to write the final vector.
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$\overrightarrow{PQ} = -3\mathbf{i} + 2\mathbf{j}$

Exam tip:

Always double check the direction of the vector: if you are calculating the vector from P to Q, you subtract P's coordinates from Q's, not the other way around, which is a common sign error.

2. Calculating Resolved Parts of a Vector★★★☆☆⏱ 5 min

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📘 Definition

Resolved parts of a vector

$V_{x} = V cos θ$ , $V_{y} = V sin θ$

The resolved parts of a vector of magnitude $V$ at an angle $θ$ to the positive x-axis are the horizontal (x-axis) and vertical (y-axis) scalar components of the vector.

The angle $θ$ is always measured anticlockwise from the positive horizontal x-axis unless stated otherwise. If the vector points below the x-axis, $θ$ will be negative, or you can add a negative sign to the vertical component directly. If the angle is given relative to the y-axis, swap cosine and sine for the respective components.

📐 Worked Example

A force vector has magnitude 12 N and acts at an angle of 40° above the positive x-axis. Find its horizontal and vertical resolved parts, and write the vector in i-j form. Give your answers to 2 decimal places.

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Step 1: Identify the given values: magnitude $V = 12$ , angle $θ = 4 0^{\circ}$ .
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Step 2: Calculate the horizontal resolved part using $V_{x} = V cos θ$ .
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$V_x = 12 \times \cos(40^\circ) = 12 \times 0.7660 = 9.19 \text{ (2 d.p.)}$
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Step 3: Calculate the vertical resolved part using $V_{y} = V sin θ$ .
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$V_y = 12 \times \sin(40^\circ) = 12 \times 0.6428 = 7.71 \text{ (2 d.p.)}$
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Step 4: Write the vector in component form.
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$9.19\mathbf{i} + 7.71\mathbf{j} \text{ N}$

3. Combining Vector Components★★★☆☆⏱ 3 min

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When you are given two or more vectors, you can find the resultant vector by adding their respective i and j components separately. This skill is the foundation for all more complex vector calculations you will encounter in the 4PM1 exam.

📐 Worked Example

Two vectors are given: $a = 4 i - 2 j$ and $b = - 1 i + 5 j$ . Find the resultant vector $a + b$ in component form.

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Step 1: Add the i components of both vectors.
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$\text{Total i component} = 4 + (-1) = 3$
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Step 2: Add the j components of both vectors.
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$\text{Total j component} = -2 + 5 = 3$
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Step 3: Write the resultant vector.
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$\mathbf{a} + \mathbf{b} = 3\mathbf{i} + 3\mathbf{j}$

✓ Quick check

A vector of magnitude 10 acts at 30° below the positive x-axis. What is its j component?
- $5 j$
- $- 5 j$
- $53 j$
- $- 53 j$
Reveal answer
1 —
The angle below the x-axis means the vertical component is negative: $10 \times sin (- 3 0^{\circ}) = - 10 \times 0.5 = - 5$ , so the j component is $- 5 j$ .

4. Common Pitfalls

Wrong move:

Subtracting end coordinates from start coordinates when calculating a vector between two points

Why:

This reverses the direction of the vector, leading to incorrect sign values for both i and j components

Correct move:

Always subtract the coordinates of the starting point from the coordinates of the end point of the vector

Wrong move:

Using sine for the horizontal component and cosine for the vertical component when angle is measured from the x-axis

Why:

The horizontal component is adjacent to the angle with the x-axis so uses cosine, while the vertical is opposite so uses sine

Correct move:

Remember: adjacent = hypotenuse × cos(theta) for horizontal x-component, opposite = hypotenuse × sin(theta) for vertical y-component when angle is measured from x-axis

Wrong move:

Forgetting to add a negative sign to components when the vector points left or down relative to the positive axes

Why:

This leads to overestimating or misplacing the direction of the resultant vector in calculations

Correct move:

Assign negative values to i components for leftward displacements, and negative j components for downward displacements

Wrong move:

Using radian mode on the calculator when calculating cosine and sine of angles given in degrees

Why:

All Edexcel 4PM1 vector questions use angles in degrees, so radian mode will give incorrect component values

Correct move:

Always check your calculator is set to degree mode before starting any vector component calculation in the exam

5. Quick Reference Cheatsheet

Task	Formula / Rule	Example
Write vector between (x1,y1) and (x2,y2) in i-j form	$(x 2 - x 1) i + (y 2 - y 1) j$	From (1,2) to (4,6): $3 i + 4 j$
Find horizontal resolved part of magnitude V at angle θ to +x axis	$V_{x} = V cos θ$	V=20, θ=30°: $V_{x} = 20 cos 30° = 103$
Find vertical resolved part of magnitude V at angle θ to +x axis	$V_{y} = V sin θ$	V=20, θ=30°: $V_{y} = 20 sin 30° = 10$
Add two vectors in component form	Add i components, add j components separately	$(2 i + 3 j) + (4 i - 1 j) = 6 i + 2 j$

6. Frequently Asked

Do I use degrees for angles in vector component questions?

Yes, all angles for resolved parts in the Edexcel 4PM1 exam are given in degrees unless explicitly stated otherwise. Always confirm your calculator is set to degree mode before starting calculations.

What if the vector is angled below the positive x-axis?

The vertical resolved part will be negative, so add a minus sign to your j component, or use the negative value of the angle in your sine calculation to get the correct sign automatically.

What's Next

Now that you can express vectors in i-j component form and calculate their resolved parts, you are ready to move on to calculating the magnitude of vectors, the next core skill in the vectors unit for Edexcel IGCSE Further Pure Maths. You will also use component form extensively when working with position vectors, and later when solving problems involving forces and kinematics using vectors. Mastering component calculations now will save you time and reduce errors in more complex vector problems that appear frequently in both Paper 1 and Paper 2 of the 4PM1 exam.