[1.3/1.4] Starting with the definition 1 in. = 2.54 cm, find the number of (a) kilometers …

Units, Physical Quantities, and Vectors

Physical Quantities

Physics deals with physical quantities, which describe properties or phenomena that can be measured. These quantities fall into two main categories:

• Scalars: Quantities with only magnitude (e.g., mass, time, temperature).
• Vectors: Quantities with both magnitude and direction (e.g., velocity, force, displacement).

To study and compare these quantities, we use a standard system of measurement called the International System of Units (SI).

Units and Measurement

Every physical quantity is expressed as a combination of a number and a unit. For example:

   

The SI system includes seven base units:

• Length: meter ()
• Mass: kilogram ()
• Time: second ()
• Electric current: ampere ()
• Temperature: kelvin ()
• Amount of substance: mole ()
• Luminous intensity: candela ()

Derived quantities, like velocity () or force (), are built from these base units.

Vectors

Vectors are crucial in physics because they allow us to describe physical phenomena involving direction. A vector is represented by an arrow:

• The length of the arrow indicates the vector’s magnitude.
• The direction of the arrow indicates the vector’s direction.

A vector can be expressed mathematically as:

   

Here:

• : Components of the vector along the x, y, and z axes.
• : Unit vectors along the x, y, and z directions.

Operations with Vectors

The two most common operations with vectors are:

• Addition: Combine vectors by placing them head-to-tail or by adding their components.
• Dot Product: Calculate the projection of one vector onto another:

     

• Cross Product: Determine a vector perpendicular to two given vectors:

     

Key Takeaways

Units standardize measurements, physical quantities describe the world, and vectors help us analyze direction-dependent phenomena. Understanding these basics lays the foundation for deeper exploration in physics.

[1.5] Uncertainty and Significant Figures. With a wooden ruler, you measure the length …

With a wooden ruler, you measure the length of a rectangular piece of sheet metal to be 12 mm. With micrometer calipers, you measure the width of the rectangle to be 5.98 mm. Use the correct number of significant figures: What are – Length of the rectangular piece of sheet metal measured with a wooden ruler: 12 mm – Width of the rectangle measured with micrometer calipers: 5.98 mm

[1.6] Estimates and Orders of Magnitude – BIO: A rather ordinary middle-aged man is in the hospital

A rather ordinary middle-aged man is in the hospital for a routine checkup. The nurse writes “200” on the patient’s medical chart but forgets to include the units. Which of these quantities could the 200 plausibly represent? The patient’s:

[1.7] Vectors and Vector Addition Worked Example 1. [1] A postal employee drives a delivery …

A postal employee drives a delivery truck along the route shown. Determine the magnitude and direction of the resultant displacement by drawing a scale diagram.

[1.8] Components of Vectors.Let θ be the angle that the vector A makes with the +x-axis, measured

Let θ be the angle that the vector A makes with the +x-axis, measured counterclockwise from that axis. Find angle θ for a vector that has these components:

[1.9] Unit Vectors. In each case, find x- and y-components of vector A:

[1.10] Products of Vectors. [a] Find the scalar product of the vectors A and B given in

[a] Find the scalar product of the vectors A and B given in Exercise 1.36. [b] Find the angle between these two vectors.

[1.11] CHALLENGE PROBLEM Physics and Vectors: The football team at Enormous State University (ESU)

The football team at Enormous State University (ESU) uses vector displacements to record its plays, with the origin taken to be the position of the ball before the play starts. In a certain pass play, the receiver starts at (1.0î − 5.0ĵ), where the units are yards, î is to the right, and ĵ is downfield. Subsequent displacements of the receiver are +9.0î (he is in motion before the snap), +11.0ĵ (breaks downfield), −6.0î + 4.0ĵ (zigs), and +12.0î + 18.0ĵ (zags). Meanwhile, the quarterback has dropped straight back to a position −7.0ĵ. How far and in which direction must the quarterback throw the ball? (Like the coach, you’ll be well advised to diagram the situation before solving this numerically.)