Vectors describe quantities with both magnitude and direction and are essential across many professional fields in Kenya, including construction project management, hospital logistics, and agricultural planning. Understanding vectors allows technicians to solve problems involving forces, displacements, and velocities in two and three dimensions. This chapter focuses on applying vector theory through operations such as addition and subtraction, and the use of position vectors to model locations and movements relevant to diverse sectors.
Vectors represent quantities with direction and magnitude, while scalars have magnitude only. In practical terms, vectors might represent displacement or force, and scalars represent temperature or cost. This section covers fundamental operations on vectors, addition and subtraction, in both two and three dimensions, enabling students to analyze real-world problems like material transport across a farm or equipment positioning in a hospital.
Vectors are quantities that have both magnitude and direction, such as velocity or force. Scalars have only magnitude, like mass or temperature.
The vector in two dimensions is typically represented as \( \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} \), where \( A_x \) and \( A_y \) are components along the x- and y-axes respectively. In three dimensions, it extends to \( \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k} \), including the z-axis.
The magnitude of a vector \( \mathbf{A} \) in two dimensions is given by:
$$ |\mathbf{A}| = \sqrt{A_x^2 + A_y^2} $$
In three dimensions:
$$ |\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} $$
Example 1: A delivery drone at a Nairobi hospital moves 3 km east and 4 km north. Find the magnitude of its displacement vector.
Given: \( A_x = 3\, \text{km}, A_y = 4\, \text{km} \)
$$ |\mathbf{A}| = \sqrt{3^2 + 4^2} $$
$$ = \sqrt{9 + 16} $$
$$ = \sqrt{25} $$
Answer: 5 km
Example 2: A security camera in a retail store is positioned at \( \mathbf{B} = 2 \mathbf{i} + 5 \mathbf{j} + 4 \mathbf{k} \) meters from the origin. Calculate its distance from the origin.
Given: \( B_x = 2\, m, B_y = 5\, m, B_z = 4\, m \)
$$ |\mathbf{B}| = \sqrt{2^2 + 5^2 + 4^2} $$
$$ = \sqrt{4 + 25 + 16} $$
$$ = \sqrt{45} $$
$$ = 6.708\, m $$
Answer: 6.708 m
Example 3: A cooperative farm worker walks 6 meters south and then 8 meters west. Represent the displacement vector and find its magnitude.
Given: \( A_x = -8\, m, A_y = -6\, m \) (south and west taken as negative directions)
$$ |\mathbf{A}| = \sqrt{(-8)^2 + (-6)^2} $$
$$ = \sqrt{64 + 36} $$
$$ = \sqrt{100} $$
Answer: 10 m
Vector addition combines two or more vectors to find a resultant vector. In two dimensions, if \( \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} \) and \( \mathbf{B} = B_x \mathbf{i} + B_y \mathbf{j} \), then:
$$ \mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \mathbf{i} + (A_y + B_y) \mathbf{j} $$
In three dimensions:
$$ \mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \mathbf{i} + (A_y + B_y) \mathbf{j} + (A_z + B_z) \mathbf{k} $$
Example 1: In a county government office, a worker moves 5 m east and 3 m north, then 2 m west and 4 m north. Find the total displacement vector and magnitude.
Given:
\( \mathbf{A} = 5 \mathbf{i} + 3 \mathbf{j} \, m \)
\( \mathbf{B} = -2 \mathbf{i} + 4 \mathbf{j} \, m \)
$$ \mathbf{R} = (5 + (-2)) \mathbf{i} + (3 + 4) \mathbf{j} $$
$$ = 3 \mathbf{i} + 7 \mathbf{j} $$
Magnitude:
$$ |\mathbf{R}| = \sqrt{3^2 + 7^2} $$
$$ = \sqrt{9 + 49} $$
$$ = \sqrt{58} $$
$$ = 7.615\, m $$
Answer: \(\mathbf{R} = 3 \mathbf{i} + 7 \mathbf{j}\) m, magnitude = 7.615 m
Example 2: A drone delivering medical supplies to a rural hospital moves 10 m north, 5 m east, then ascends 3 m vertically. Find the resultant displacement vector in 3D.
Given:
\( \mathbf{A} = 10 \mathbf{j} + 5 \mathbf{i} + 0 \mathbf{k} \, m \)
\( \mathbf{B} = 0 \mathbf{i} + 0 \mathbf{j} + 3 \mathbf{k} \, m \)
$$ \mathbf{R} = (5 + 0) \mathbf{i} + (10 + 0) \mathbf{j} + (0 + 3) \mathbf{k} $$
$$ = 5 \mathbf{i} + 10 \mathbf{j} + 3 \mathbf{k} $$
Magnitude:
$$ |\mathbf{R}| = \sqrt{5^2 + 10^2 + 3^2} $$
$$ = \sqrt{25 + 100 + 9} $$
$$ = \sqrt{134} $$
$$ = 11.575\, m $$
Answer: \(\mathbf{R} = 5 \mathbf{i} + 10 \mathbf{j} + 3 \mathbf{k}\) m, magnitude = 11.575 m
Example 3: A hotel porter moves luggage 7 m west, 4 m south, and 2 m up to a guest room. Another porter moves luggage 3 m east, 6 m north, and 1 m down. Find the resultant vector and magnitude.
Given:
\( \mathbf{A} = -7 \mathbf{i} - 4 \mathbf{j} + 2 \mathbf{k} \, m \)
\( \mathbf{B} = 3 \mathbf{i} + 6 \mathbf{j} - 1 \mathbf{k} \, m \)
$$ \mathbf{R} = (-7 + 3) \mathbf{i} + (-4 + 6) \mathbf{j} + (2 + (-1)) \mathbf{k} $$
$$ = -4 \mathbf{i} + 2 \mathbf{j} + 1 \mathbf{k} $$
Magnitude:
$$ |\mathbf{R}| = \sqrt{(-4)^2 + 2^2 + 1^2} $$
$$ = \sqrt{16 + 4 + 1} $$
$$ = \sqrt{21} $$
$$ = 4.583\, m $$
Answer: \(\mathbf{R} = -4 \mathbf{i} + 2 \mathbf{j} + 1 \mathbf{k}\) m, magnitude = 4.583 m
Vector subtraction finds the difference between two vectors, often representing displacement between points. For vectors \( \mathbf{A} \) and \( \mathbf{B} \):
$$ \mathbf{D} = \mathbf{A} - \mathbf{B} = (A_x - B_x) \mathbf{i} + (A_y - B_y) \mathbf{j} $$
In three dimensions:
$$ \mathbf{D} = (A_x - B_x) \mathbf{i} + (A_y - B_y) \mathbf{j} + (A_z - B_z) \mathbf{k} $$
Example 1: A farmer walks from point A at (3, 2) meters to point B at (7, 5) meters on the farm. Find the displacement vector from A to B.
Given:
\( \mathbf{A} = 3 \mathbf{i} + 2 \mathbf{j} \, m \)
\( \mathbf{B} = 7 \mathbf{i} + 5 \mathbf{j} \, m \)
$$ \mathbf{D} = \mathbf{B} - \mathbf{A} = (7 - 3) \mathbf{i} + (5 - 2) \mathbf{j} $$
$$ = 4 \mathbf{i} + 3 \mathbf{j} $$
Magnitude:
$$ |\mathbf{D}| = \sqrt{4^2 + 3^2} $$
$$ = \sqrt{16 + 9} $$
$$ = \sqrt{25} $$
Answer: \(\mathbf{D} = 4 \mathbf{i} + 3 \mathbf{j}\) m, magnitude = 5 m
Example 2: A vehicle in a county government parking lot moves from position \( \mathbf{P} = 12 \mathbf{i} + 8 \mathbf{j} + 0 \mathbf{k} \, m \) to \( \mathbf{Q} = 7 \mathbf{i} + 3 \mathbf{j} + 2 \mathbf{k} \, m \). Find the displacement vector \( \mathbf{D} = \mathbf{Q} - \mathbf{P} \).
Given:
\( \mathbf{P} = 12 \mathbf{i} + 8 \mathbf{j} + 0 \mathbf{k} \, m \)
\( \mathbf{Q} = 7 \mathbf{i} + 3 \mathbf{j} + 2 \mathbf{k} \, m \)
$$ \mathbf{D} = (7 - 12) \mathbf{i} + (3 - 8) \mathbf{j} + (2 - 0) \mathbf{k} $$
$$ = -5 \mathbf{i} - 5 \mathbf{j} + 2 \mathbf{k} $$
Magnitude:
$$ |\mathbf{D}| = \sqrt{(-5)^2 + (-5)^2 + 2^2} $$
$$ = \sqrt{25 + 25 + 4} $$
$$ = \sqrt{54} $$
$$ = 7.348\, m $$
Answer: \(\mathbf{D} = -5 \mathbf{i} - 5 \mathbf{j} + 2 \mathbf{k}\) m, magnitude = 7.348 m
Example 3: A technician moves equipment from point A (4, 6, 3) meters to point B (9, 2, 7) meters in a hospital. Find the displacement vector and its magnitude.
Given:
\( \mathbf{A} = 4 \mathbf{i} + 6 \mathbf{j} + 3 \mathbf{k} \, m \)
\( \mathbf{B} = 9 \mathbf{i} + 2 \mathbf{j} + 7 \mathbf{k} \, m \)
$$ \mathbf{D} = (9 - 4) \mathbf{i} + (2 - 6) \mathbf{j} + (7 - 3) \mathbf{k} $$
$$ = 5 \mathbf{i} - 4 \mathbf{j} + 4 \mathbf{k} $$
Magnitude:
$$ |\mathbf{D}| = \sqrt{5^2 + (-4)^2 + 4^2} $$
$$ = \sqrt{25 + 16 + 16} $$
$$ = \sqrt{57} $$
$$ = 7.550\, m $$
Answer: \(\mathbf{D} = 5 \mathbf{i} - 4 \mathbf{j} + 4 \mathbf{k}\) m, magnitude = 7.550 m
Multiplying a vector by a scalar changes its magnitude but not its direction unless the scalar is negative, which reverses direction. For a vector \( \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k} \) and scalar \( k \):
$$ k \mathbf{A} = k A_x \mathbf{i} + k A_y \mathbf{j} + k A_z \mathbf{k} $$
Example 1: A technician at a county referral hospital moves a cart with displacement vector \( \mathbf{A} = 4 \mathbf{i} + 3 \mathbf{j} \) meters. Find the displacement after 3 trips.
Given:
\( \mathbf{A} = 4 \mathbf{i} + 3 \mathbf{j} \, m \)
\( k = 3 \)
$$ k \mathbf{A} = 3 \times 4 \mathbf{i} + 3 \times 3 \mathbf{j} $$
$$ = 12 \mathbf{i} + 9 \mathbf{j} $$
Magnitude:
$$ |k \mathbf{A}| = 3 \times \sqrt{4^2 + 3^2} $$
$$ = 3 \times 5 $$
$$ = 15\, m $$
Answer: \( \mathbf{R} = 12 \mathbf{i} + 9 \mathbf{j} \) m, magnitude = 15 m
Example 2: A drone moves with velocity vector \( \mathbf{V} = 2 \mathbf{i} + 5 \mathbf{j} + 1 \mathbf{k} \) m/s. Find the velocity vector after doubling speed.
Given:
\( \mathbf{V} = 2 \mathbf{i} + 5 \mathbf{j} + 1 \mathbf{k} \, m/s \)
\( k = 2 \)
$$ k \mathbf{V} = 2 \times 2 \mathbf{i} + 2 \times 5 \mathbf{j} + 2 \times 1 \mathbf{k} $$
$$ = 4 \mathbf{i} + 10 \mathbf{j} + 2 \mathbf{k} $$
Magnitude:
$$ |k \mathbf{V}| = 2 \times \sqrt{2^2 + 5^2 + 1^2} $$
$$ = 2 \times \sqrt{4 + 25 + 1} $$
$$ = 2 \times \sqrt{30} $$
$$ = 2 \times 5.477 $$
$$ = 10.954\, m/s $$
Answer: \( \mathbf{V} = 4 \mathbf{i} + 10 \mathbf{j} + 2 \mathbf{k} \) m/s, magnitude = 10.954 m/s
Example 3: A cooperative farm worker applies a force vector \( \mathbf{F} = 6 \mathbf{i} - 2 \mathbf{j} \) N. Find the force vector when the force magnitude is halved.
Given:
\( \mathbf{F} = 6 \mathbf{i} - 2 \mathbf{j} \, N \)
\( k = 0.5 \)
$$ k \mathbf{F} = 0.5 \times 6 \mathbf{i} + 0.5 \times (-2) \mathbf{j} $$
$$ = 3 \mathbf{i} - 1 \mathbf{j} $$
Magnitude:
$$ |k \mathbf{F}| = 0.5 \times \sqrt{6^2 + (-2)^2} $$
$$ = 0.5 \times \sqrt{36 + 4} $$
$$ = 0.5 \times \sqrt{40} $$
$$ = 0.5 \times 6.325 $$
$$ = 3.1625\, N $$
Answer: \( \mathbf{F} = 3 \mathbf{i} - 1 \mathbf{j} \) N, magnitude = 3.163 N
A security guard walks 8 m east and 6 m north. Find the magnitude of the displacement vector. (3 marks)
A technician moves equipment 4 m west, 3 m south, and 2 m up. Another technician moves it 1 m east, 5 m north, and 3 m down. Find the resultant displacement vector and magnitude. (5 marks)
A delivery person moves from point A (2, 4, 1) meters to point B (7, 1, 5) meters. Find the displacement vector and its magnitude. (4 marks)
A farm worker moves 5 m north, then 12 m east. Find the total displacement vector and its magnitude. (4 marks)
A drone moves with velocity vector \( \mathbf{V} = 3 \mathbf{i} + 4 \mathbf{j} \) m/s. Find the velocity vector after tripling the speed and its magnitude. (4 marks)
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Create a free accountThis chapter covered the fundamental concepts of vectors and scalars in two and three dimensions, emphasizing their distinct properties and how to perform addition and subtraction operations on vectors. It introduced position vectors as a way to represent points in space relative to an origin. The resolution of vectors into components was explained to facilitate analysis in multiple directions. The chapter further explored scalar and vector products, detailing their calculation and significance in engineering contexts. The gradient was presented as a measure of how a scalar field changes in space, followed by the curl which describes the rotation of a vector field. Finally, divergence was examined as an indicator of the magnitude of a source or sink at a given point within a vector field. Together, these topics provide a comprehensive foundation for applying vector theory in engineering problems.
A delivery truck at a county hospital moves 3 km east and then 4 km north. Find the resultant displacement vector magnitude. (2 marks)
A position vector of a water tank in a hotel grounds is given by \(\mathbf{r} = 5\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) metres. Find the magnitude of \(\mathbf{r}\). (2 marks)