Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Vectors in two dimension
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Vectors in two dimension
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: Further Mathematics
Class: Senior Secondary 1
Term: 3rd Term
Week: 2
Theme: Vectors
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
State the difference between vectors and scalars Perform simple operations on vectors Determine the sum, difference of any combination of vectors lying in a plane Resolve a vector in a given direction Define scalar (dot) product and its application
Scalar Quantity: A physical quantity that has only magnitude (size). It is completely described by a numerical value and a unit.
Examples: Mass (e.g., 50 kg of yam), temperature (e.g., 30°C in Lagos), distance (e.g., 10 km from Ibadan to Oyo), speed (e.g., a car traveling at 80 km/h), time, energy, volume.
Vector Quantity: A physical quantity that has both magnitude and direction. It requires both a numerical value, a unit, and a specified direction for its complete description.
Examples: Displacement (e.g., 10 km North-East from Ibadan), velocity (e.g., a car traveling at 80 km/h East), force (e.g., 100 N applied downwards), acceleration, momentum, weight.
Representation of Vectors: Vectors are typically represented by: Bold lowercase letters: e.g., a, b, r.
Letters with an arrow above: e.g., $\vec{a}$, $\vec{b}$, $\vec{r}$.
Directed line segments: An arrowed line segment where the length represents the magnitude and the arrowhead indicates the direction. If a vector starts at point A and ends at point B, it is denoted as $\vec{AB}$.
Column Vectors: In a Cartesian coordinate system, a vector from the origin O to a point P(x, y) can be written as $\vec{OP} = \begin{pmatrix} x \\ y \end{pmatrix}$. This is also called a position vector.
Unit Vector Notation: A vector can also be expressed in terms of unit vectors i and j, where i is a unit vector along the positive x-axis and j is a unit vector along the positive y-axis. So, $\vec{r} = x\mathbf{i} + y\mathbf{j}$. Two vectors are equal if and only if they have the same magnitude and the same direction. Their starting points do not have to coincide. If $\vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}$, then $\vec{a} = \vec{b}$ implies $x_1 = x_2$ and $y_1 = y_2$. The magnitude of a vector $\vec{r} = \begin{pmatrix} x \\ y \end{pmatrix}$ or $\vec{r} = x\mathbf{i} + y\mathbf{j}$ is its length, denoted by $|\vec{r}|$ or $r$. Using Pythagoras' theorem, it is calculated as: $|\vec{r}| = \sqrt{x^2 + y^2}$ Example 2.3.1: A surveyor measures the displacement from point A to point B as $\vec{AB} = 3\mathbf{i} + 4\mathbf{j}$ km. Calculate the magnitude of this displacement.
Solution: Given $\vec{AB} = 3\mathbf{i} + 4\mathbf{j}$, the x-component is 3 and the y-component is 4. $|\vec{AB}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ km. The direction of a vector is usually specified by the angle it makes with the positive x-axis, measured anti-clockwise. If $\vec{r} = \begin{pmatrix} x \\ y \end{pmatrix}$ and $\theta$ is the angle with the positive x-axis, then $\tan \theta = \frac{y}{x}$. The quadrant of the vector determines the actual angle: Quadrant 1 (x>0, y>0): $\theta = \arctan(\frac{y}{x})$ Quadrant 2 (x0): $\theta = 180^\circ - \arctan(|\frac{y}{x}|)$ Quadrant 3 (x0, y 0$, the direction remains the same. If $k x-axis | / | / | / 120 N (F) | / | / | / 25 deg (angle below horizontal) | / | V | F_y (downwards) | V ``` (Students should draw this showing the force vector in the 4th quadrant, making 25° with the positive x-axis, pointing downwards.) b) Given magnitude $|\vec{F}| = 120$ N. The angle with the positive x-axis is $-25^\circ$ or $360^\circ - 25^\circ = 335^\circ$. Horizontal component $F_x = |\vec{F}|\cos\theta = 120 \cos(-25^\circ) = 120 \cos(25^\circ)$. $F_x = 120 \times 0.9063 \approx 108.76$ N. Vertical component $F_y = |\vec{F}|\sin\theta = 120 \sin(-25^\circ) = -120 \sin(25^\circ)$. $F_y = -120 \times 0.4226 \approx -50.71$
N. So, the force vector is approximately $\vec{F} = 108.76\mathbf{i} - 50.71\mathbf{j}$
N. Commentary: This problem introduces resolving vectors with an angle below the horizontal, requiring careful attention to the sign of the vertical component. The diagram reinforces understanding.
Question 4 (Objective 5): Two engineering students are analyzing forces on a cantilever beam. They model two forces as $\vec{F_1} = 6\mathbf{i} + 8\mathbf{j}$ N and $\vec{F_2} = -4\mathbf{i} + 3\mathbf{j}$ N. a) Calculate the scalar product $\vec{F_1} \cdot \vec{F_2}$. b) Determine if these two forces are perpendicular.
Solution 4: a) $\vec{F_1} \cdot \vec{F_2} = (6)(-4) + (8)(3)$ $= -24 + 24 = 0$. b) Since $\vec{F_1} \cdot \vec{F_2} = 0$, the two forces $\vec{F_1}$ and $\vec{F_2}$ are perpendicular to each other.
Commentary: This question directly tests the application of the dot product to determine perpendicularity, a key concept for understanding force interactions in engineering. ---
Navigation and Transportation: Vectors are fundamental to navigation systems used by `Okada` riders, commercial `danfo` drivers, pilots, and sea captains.
Application: A ferry boat crossing the Niger River needs to account for its own velocity relative to the water and the velocity of the river current. Vectors allow engineers to calculate the resultant velocity and direction the boat must steer to reach its destination accurately, avoiding being swept downstream or hitting sandbanks. Similarly, pilots use vectors to account for wind speed and direction to maintain their intended course and speed.
Civil Engineering and Construction: When designing structures like bridges, buildings, or even simple roof trusses in Nigerian homes, engineers use vectors to analyze forces.
Application: The weight of a roof, wind pressure, and other loads create forces acting on various parts of a structure. By resolving these forces into their horizontal and vertical components, engineers can determine the stress on beams and columns, ensuring the structure is stable and safe, able to withstand environmental factors like strong winds during the rainy season.
Sports and Athletics: Vectors play a role in analyzing motion and forces in various sports popular in Nigeria.
Application: In football (soccer), the trajectory of a kicked ball is determined by the initial velocity (magnitude and direction) and gravitational force. Coaches can use vector principles to analyze the force and angle required for a player to shoot for goal or pass the ball effectively. Similarly, in track and field, the physics of a javelin throw or long jump involves understanding the optimum angle and initial velocity (vector quantities) for maximum distance. ---