Vectors are fundamental in analytical chemistry technology for describing quantities that have both magnitude and direction, such as forces in laboratory apparatus, displacement of particles, or velocity of fluid flow in separation processes. Mastery of vectors enables professionals to analyze multicomponent systems, resolve forces, and understand spatial relationships critical in precise measurements and instrumentation. This chapter develops skills to represent vectors graphically and perform vector operations essential for applied analytical chemistry tasks.

7.1 Vectors and scalar quantities in two dimensions

Vector quantities have both magnitude and direction, unlike scalar quantities which only have magnitude. In analytical chemistry laboratories, understanding this distinction is crucial when measuring quantities like velocity of fluid flow (vector) versus temperature (scalar). This section defines these quantities and shows how to draw vectors accurately in two dimensions.

7.1.1 Definitions of vector and scalar quantities

A scalar quantity is a physical measurement described by magnitude alone, without direction. Examples include temperature, mass, volume, and concentration. A vector quantity has both magnitude and direction, such as displacement, velocity, force, and acceleration.

For a vector \(\vec{A}\), its magnitude is denoted \(|\vec{A}|\) and its direction is often given by an angle \(\theta\) relative to a reference axis.

Definitions Formula

For a vector \(\vec{A}\) in two dimensions with components \(A_x\) and \(A_y\):

$$|\vec{A}| = \sqrt{A_x^2 + A_y^2}$$

$$\theta = \tan^{-1}(\frac{A_y}{A_x})$$

Worked Examples

Example 1: An ion moves 3 m east and 4 m north in a laboratory. Find the magnitude and direction of the displacement vector.

Given: \(A_x = 3\,m\), \(A_y = 4\,m\)

$$|\vec{A}| = \sqrt{3^2 + 4^2}$$

$$= \sqrt{9 + 16}$$

$$= \sqrt{25}$$

$$= 5\,m$$

$$\theta = \tan^{-1}(\frac{4}{3})$$

$$= \tan^{-1}(1.333)$$

$$= 53.13^\circ$$

Answer: Magnitude = 5 m, Direction = 53.13° north of east

Example 2: A chemical sample moves with velocity components \(v_x = 6\,ms^{-1}\) and \(v_y = 8\,ms^{-1}\). Find the magnitude and direction of velocity.

Given: \(v_x = 6\,ms^{-1}\), \(v_y = 8\,ms^{-1}\)

$$|\vec{v}| = \sqrt{6^2 + 8^2}$$

$$= \sqrt{36 + 64}$$

$$= \sqrt{100}$$

$$= 10\,ms^{-1}$$

$$\theta = \tan^{-1}(\frac{8}{6})$$

$$= \tan^{-1}(1.333)$$

$$= 53.13^\circ$$

Answer: Velocity magnitude = 10 ms\(^{-1}\), direction = 53.13° north of east

Example 3: A force acts on a sample with components \(F_x = -5\,N\) (west) and \(F_y = 12\,N\) (north). Find magnitude and direction relative to east.

Given: \(F_x = -5\,N\), \(F_y = 12\,N\)

$$|\vec{F}| = \sqrt{(-5)^2 + 12^2}$$

$$= \sqrt{25 + 144}$$

$$= \sqrt{169}$$

$$= 13\,N$$

$$\theta = \tan^{-1}(\frac{12}{-5})$$

$$= \tan^{-1}(-2.4)$$

Since \(F_x < 0\) and \(F_y > 0\), vector lies in second quadrant.

$$\theta = 180^\circ - 67.38^\circ = 112.62^\circ$$

Answer: Magnitude = 13 N, direction = 112.62° from east (northwest)

7.1.2 Drawing a vector

Drawing vectors accurately is essential for visualizing directions and magnitudes in experimental setups. A vector is represented by an arrow where the length is proportional to magnitude and the arrowhead points in the direction.

Steps to draw a vector:

1. Choose scale (e.g., 1 cm = 1 unit).
2. Draw a reference axis (usually x-axis horizontal).
3. Measure the length proportional to the vector magnitude along the direction angle.
4. Draw an arrowhead indicating direction.
5. Label the vector with magnitude and angle.

Worked Examples

Example 1: Draw a vector of magnitude 5 units at an angle 30° from the x-axis using scale 1 cm = 1 unit.

Given: Magnitude = 5 units, Angle = 30°, Scale = 1 cm/unit

• Length = 5 cm
• Starting at origin, measure 5 cm at 30° angle from x-axis.
• Draw an arrowhead at the end.
• Label vector as \( \vec{A} = 5 \) units, \(30^\circ\).

Example 2: A force vector of 8 N acts at 120° from the x-axis. Draw using scale 1 cm = 2 N.

Given: Magnitude = 8 N, Angle = 120°, Scale = 1 cm = 2 N

• Length = \( \frac{8}{2} = 4 \) cm
• From origin, measure 4 cm at 120°
• Draw arrowhead and label \( \vec{F} = 8 \, N, 120^\circ \).

Example 3: A velocity vector of 10 ms\(^{-1}\) at 225° from x-axis. Draw with scale 1 cm = 2 ms\(^{-1}\).

Given: Magnitude = 10 ms\(^{-1}\), Angle = 225°, Scale = 1 cm = 2 ms\(^{-1}\)

• Length = \( \frac{10}{2} = 5 \) cm
• From origin, measure 5 cm at 225°
• Draw arrowhead and label \( \vec{v} = 10\, ms^{-1}, 225^\circ \).

Chapter Summary

This chapter introduced the fundamental concepts of vectors and scalar quantities in two dimensions, explaining their definitions and how to represent vectors graphically. It covered vector operations including addition, subtraction, and scalar multiplication, demonstrating how these are performed and their significance in problem-solving. The chapter then explored the position of vectors with a focus on calculating the modulus, or magnitude, which quantifies the length of a vector. Finally, it examined the resolution of a vector into its components along specified directions, enabling easier analysis of vector quantities in practical applications. Throughout, the chapter provided a structured approach to understanding and manipulating vectors in two-dimensional contexts relevant to science and engineering.