By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world problems that involve vectors and prepare you for more advanced math in your trade.
Vectors are fundamental in science laboratory technology, where precise measurement and representation of physical quantities such as force, velocity, and displacement are crucial. Mastery of vectors enables laboratory technologists to analyze and interpret data involving direction and magnitude, which is essential in experiments and instrumentation handling. This chapter introduces vectors and their properties, distinguishing them from scalar quantities, to build a solid foundation for vector operations in scientific contexts.
Vectors describe quantities that possess both magnitude and direction, essential for representing physical phenomena in laboratory settings. Understanding vectors and their difference from scalars allows technologists to perform accurate calculations involving forces, velocities, and other directional quantities.
Vectors are quantities characterized by both magnitude and direction. Unlike scalar quantities, which have only magnitude, vectors must be represented graphically or algebraically to include direction information. The fundamental representation of a vector is an arrow where the length corresponds to magnitude and the arrowhead indicates direction.
The mathematical notation for a vector is typically a letter with an arrow on top, for example, \(\vec{A}\). The magnitude of a vector \(\vec{A}\) is denoted by \(|\vec{A}|\) or simply \(A\).
$$ \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} $$
where \(A_x, A_y, A_z\) are the components of the vector along the x, y, and z axes respectively, and \(\hat{i}, \hat{j}, \hat{k}\) are the unit vectors in those directions.
Example 1: A force vector \(\vec{F}\) has components \(F_x = 3\, \text{N}\) and \(F_y = 4\, \text{N}\). Find the magnitude of \(\vec{F}\).
Given: \(F_x = 3\, \text{N}, F_y = 4\, \text{N}\)
$$ |\vec{F}| = \sqrt{F_x^2 + F_y^2} $$
$$ = \sqrt{3^2 + 4^2} $$
$$ = \sqrt{9 + 16} = \sqrt{25} $$
Answer: 5 N
Example 2: A displacement vector \(\vec{D}\) has components \(D_x = 5\, \text{m}, D_y = 12\, \text{m}\). Calculate the magnitude and direction angle \(\theta\) with respect to the x-axis.
Given: \(D_x = 5\, \text{m}, D_y = 12\, \text{m}\)
$$ |\vec{D}| = \sqrt{D_x^2 + D_y^2} $$
$$ = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} $$
$$ = 13\, \text{m} $$
$$ \theta = \tan^{-1} \frac{D_y}{D_x} = \tan^{-1} \frac{12}{5} $$
$$ = \tan^{-1} 2.4 = 67.38^\circ $$
Answer: Magnitude = 13 m, Direction = 67.38°
Example 3: A velocity vector \(\vec{v}\) in three dimensions has components \(v_x = 2\, \text{m/s}, v_y = -3\, \text{m/s}, v_z = 6\, \text{m/s}\). Find the magnitude of \(\vec{v}\).
Given: \(v_x = 2\, \text{m/s}, v_y = -3\, \text{m/s}, v_z = 6\, \text{m/s}\)
$$ |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} $$
$$ = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} $$
Answer: 7 m/s
Vector quantities have both magnitude and direction, making them vital in laboratory measurements involving spatial orientation or directional effects. Common vector quantities encountered in laboratory technology include force, displacement, velocity, acceleration, and electric field.
The general form for expressing a vector quantity is through its components along coordinate axes or by magnitude and direction.
$$ \vec{V} = V_x \hat{i} + V_y \hat{j} + V_z \hat{k} $$
Example 1: A force vector \(\vec{F}\) in a chemical experiment has components \(F_x = 6\, \text{N}\) and \(F_y = 8\, \text{N}\). Find the magnitude and direction angle.
Given: \(F_x = 6\, \text{N}, F_y = 8\, \text{N}\)
$$ |\vec{F}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} $$
$$ = 10\, \text{N} $$
$$ \theta = \tan^{-1} \frac{8}{6} = \tan^{-1} 1.333 = 53.13^\circ $$
Answer: Magnitude = 10 N, Direction = 53.13°
Example 2: In a laboratory setup, a particle moves with velocity components \(v_x = 4\, \text{m/s}\) and \(v_y = 3\, \text{m/s}\). Calculate the speed and direction.
Given: \(v_x = 4\, \text{m/s}, v_y = 3\, \text{m/s}\)
$$ |\vec{v}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\, \text{m/s} $$
$$ \theta = \tan^{-1} \frac{3}{4} = \tan^{-1} 0.75 = 36.87^\circ $$
Answer: Speed = 5 m/s, Direction = 36.87°
Example 3: An electric field vector \(\vec{E}\) has components \(E_x = 0\, \text{N/C}, E_y = 5\, \text{N/C}, E_z = 12\, \text{N/C}\). Find its magnitude.
Given: \(E_x = 0, E_y = 5, E_z = 12\, \text{N/C}\)
$$ |\vec{E}| = \sqrt{0^2 + 5^2 + 12^2} = \sqrt{0 + 25 + 144} = \sqrt{169} $$
Answer: 13 N/C
Example 4: A laboratory technician measures the acceleration vector \(\vec{a}\) with components \(a_x = -3\, \text{m/s}^2\), \(a_y = 4\, \text{m/s}^2\). Find the magnitude and direction relative to the positive x-axis.
Given: \(a_x = -3\, \text{m/s}^2, a_y = 4\, \text{m/s}^2\)
$$ |\vec{a}| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\, \text{m/s}^2 $$
$$ \theta = \tan^{-1} \frac{4}{-3} = \tan^{-1} (-1.333) = -53.13^\circ $$
Since \(a_x\) is negative and \(a_y\) positive, direction is in the second quadrant:
$$ \theta = 180^\circ - 53.13^\circ = 126.87^\circ $$
Answer: Magnitude = 5 m/s², Direction = 126.87°
Scalar quantities have magnitude only and no direction, distinguishing them from vectors. Scalars are used to measure properties such as temperature, mass, volume, and time in laboratory environments, where direction is irrelevant.
Scalar quantities are expressed as single numerical values with units.
Example 1: A laboratory sample has a mass of 250 g. Convert this mass into kilograms.
Given: Mass = 250 g
$$ \text{Mass} = \frac{250}{1000} = 0.25\, \text{kg} $$
Answer: 0.25 kg
Example 2: The temperature in a laboratory is 37°C. Convert this temperature to Kelvin.
Given: Temperature = 37°C
$$ T(K) = T(°C) + 273.15 = 37 + 273.15 = 310.15\, \text{K} $$
Answer: 310.15 K
Example 3: The volume of a liquid measured is 500 ml. Convert to liters.
Given: Volume = 500 ml
$$ V = \frac{500}{1000} = 0.5\, \text{L} $$
Answer: 0.5 L
Example 4: A stopwatch records a time interval of 125 seconds. Convert to minutes.
Given: Time = 125 s
$$ t = \frac{125}{60} = 2.0833\, \text{minutes} $$
Answer: 2.08 minutes
Example 5: The density of a solution is 1.2 g/cm³. Express this in kg/m³.
Given: Density = 1.2 g/cm³
$$ 1\, \text{g/cm}^3 = 1000\, \text{kg/m}^3 $$
$$ \rho = 1.2 \times 1000 = 1200\, \text{kg/m}^3 $$
Answer: 1200 kg/m³
A force vector in a laboratory experiment has components \(F_x = 7\, \text{N}\) and \(F_y = 24\, \text{N}\). Find the magnitude and direction angle of the force. (5 marks)
A particle moves with velocity components \(v_x = 8\, \text{m/s}\), \(v_y = -6\, \text{m/s}\). Calculate the magnitude and direction of the velocity vector. (5 marks)
The electric field vector in a test setup is \(\vec{E} = 3 \hat{i} + 4 \hat{j} + 12 \hat{k}\) N/C. Find the magnitude of \(\vec{E}\). (5 marks)
Convert a laboratory temperature of 25°C to Kelvin. (3 marks)
A liquid volume is recorded as 750 ml. Convert this volume to liters. (3 marks)
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Create a free accountThis chapter introduced the concept of vectors, defining them as quantities possessing both magnitude and direction, distinguishing them from scalar quantities which have only magnitude. It explained various vector quantities commonly encountered in science and engineering. The chapter then explored vector addition, demonstrating how vectors combine to form resultant vectors using graphical and analytical methods. Vector subtraction was covered next, showing how to determine the difference between two vectors by adding the negative of one vector to the other. The multiplication of vectors was discussed, including both scalar (dot) product and vector (cross) product, highlighting their different results and applications. Finally, the chapter addressed the position of vectors and the calculation of the modulus, or magnitude, of a vector, providing the basis for understanding vector length in space. Together, these topics build a foundation for applying vectors in various scientific and engineering problems.
A vector \(\vec{A}\) has components \(3\, \text{m}\) along the x-axis and \(4\, \text{m}\) along the y-axis. Calculate the modulus of \(\vec{A}\). (2 marks)
Two vectors \(\vec{B} = 5\, \text{m}\) east and \(\vec{C} = 12\, \text{m}\) north are acting on a sample in a laboratory. Find the magnitude of the resultant vector \(\vec{R} = \vec{B} + \vec{C}\). (3 marks)
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