Notes 1.4: Basic Vectors Physics Honors I

Objectives: Learn what a scalar is. Learn about vectors and motion in two dimensional space. Learn to break a vector into it’s components. Learn to find the magnitude of a vector. Learn to find the angle of a vector.

Active Physics Reference: Vector Definition: page 59 Vector Addition: page 168-169; page 450-453 Vector Subtraction: page 355 Vector Components: page 179-180 Using the Pythagorean Theorem: page 454

Further Learning: Crash Course Physics – Vectors - Video: https://www.youtube.com/watch?v=w3BhzYI6zXU Bozeman Science – Vectors – Video: https://www.youtube.com/watch?v=EUrMI0DIh40 What is a Vector? – Vectors – Video: https://www.youtube.com/watch?v=ml4NSzCQobk

What are vectors? https://www.youtube.com/watch?v=ml4NSzCQobk

Scalars vs. Vectors What is a scalar? A scalar is any substance that only has magnitude. What is a vector? A vector is any substance that has magnitude AND direction.

DEMO: Why is direction important? Take a scale, put 400 g on string, and measure the mass with the spring scale. What should the spring scale measure? 400 grams if calibrated correctly. Now add another spring scale, hold both scales vertically while letting the weight hang vertically as well. What do you expect the scales to read? We are getting a reading of approximately 200 grams on one and 200 grams on another. That makes sense.

DEMO: Why is direction important? Now lets hold the scales at an angle away from each other. What should you expect the scales to read? You probably expect that you should be getting a reading of 200 grams on one scale and 200 grams on the other scale. Why is that? Does that make sense? This is an example of when we have to account for the direction mathematically to make sense of a real world scenario.

Vector Components: A component of a vector is the projection of the vector on an axis; that is, the part of the vector that translates to three-dimensional space. What does that mean? Basically, we find how much of the vector is on the x-axis, how much of it is on the y-axis, how much of it is on the z-axis.

Vector Components: The process of finding the components of a vectors is called resolving the vector. So as in the image, we have a vector on an x-y plane:

Vector Components: We can break it into component by: Basically, we have a component of the vector 𝑑 on the x-axis that is called 𝑑 𝑥 , and a component of it on the y-axis that is called 𝑑 𝑦 .

Vector Components: Basic looking at the diagram below, we see that essentially the vector 𝑑 and its components form a triangle. This allows us to use trigonometry to create some basic formulas for describing a vector.

Vector Components: Recall that: 𝑐𝑜𝑠𝜃= 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒   𝑐𝑜𝑠𝜃= 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑠𝑖𝑛𝜃= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑡𝑎𝑛𝜃= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

Vector Components: We can find 𝑑 𝑥 (that is, the amount of vector 𝑑 on the x-axis) by using 𝑐𝑜𝑠𝜃

Vector Components: Similarly, we can find 𝑑 𝑦 (that is, the amount of vector 𝑑 on the y-axis) by using 𝑠𝑖𝑛𝜃.

Vector Components: Look back at the graph above, we have a triangle. Our magnitude is just the distance of the hypotenuse. So what equation can we use to find the hypotenuse? The Pythagorean theorem. So to find the magnitude of a vector, we use this formula:

Vector Components: What about the direction? How do we find it?

Example Problem 1: To run a route for a play in football, Derek the running back will run 8 yards to the left then turn right and run 10 yards up the field where he should be expecting to catch the pass from Erika the quarterback. What is the magnitude of the distance that Erika, the quarterback, must throw the ball? d = 12.8 yds At what angle must she throw the ball? 𝜃=−51°+180°=129°

Check Point 1: A car is driven 125.0 km due west, then 65.0 km due south. Determine the magnitude of the displacement vector. d =140 km Determine the direction (angle) of the displacement vector from the horizontal. θ = 27.5◦ + 180◦ = 207.5◦

Unit Vector Notation: 𝑖 =𝑥−𝑎𝑥𝑖𝑠 𝑜𝑟 𝑥−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛. Unit vector – a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to point – that is, to specify direction. 𝑖 =𝑥−𝑎𝑥𝑖𝑠 𝑜𝑟 𝑥−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛. 𝑗 =𝑦−𝑎𝑥𝑖𝑠 𝑜𝑟 𝑦−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛. 𝑘 =𝑧−𝑎𝑥𝑖𝑠 𝑜𝑟 𝑧−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛.

Unit Vector Notation: So what we can write the distance of a vector in the form of: 𝑑 = 𝑑 𝑥 𝑖 + 𝑑 𝑦 𝑗 𝑑 =(𝑑 cos 𝜃) 𝑖 +( dsin 𝜃) 𝑗 Remember that 𝑑 is the vector and d is the magnitude of that vector.

Vector Addition: Let’s say that we have vector 𝑑 1 and vector 𝑑 2 where we have: 𝑑 1 = 𝑑 𝑥1 𝑖 + 𝑑 𝑦1 𝑗 and 𝑑 2 = 𝑑 𝑥2 𝑖 + 𝑑 𝑦2 𝑗 How do we add them? 𝑑 1 + 𝑑 2 𝑑 1 + 𝑑 2 = 𝑑 𝑥1 𝑖 + 𝑑 𝑦1 𝑗 + (𝑑 𝑥2 𝑖 + 𝑑 𝑦2 𝑗 ) 𝑑 1 + 𝑑 2 = 𝑑 𝑥1 + 𝑑 𝑥2 𝑖 + (𝑑 𝑦1 + 𝑑 𝑦2 ) 𝑗

Vector Subtraction: Let’s say that we have vector 𝑑 1 and vector 𝑑 2 where we have: 𝑑 1 = 𝑑 𝑥1 𝑖 + 𝑑 𝑦1 𝑗 and 𝑑 2 = 𝑑 𝑥2 𝑖 + 𝑑 𝑦2 𝑗 How do we subtract them? 𝑑 1 − 𝑑 2 = 𝑑 1 +(− 𝑑 2 ) 𝑑 1 − 𝑑 2 = 𝑑 𝑥1 𝑖 + 𝑑 𝑦1 𝑗 − (𝑑 𝑥2 𝑖 + 𝑑 𝑦2 𝑗 ) 𝑑 1 − 𝑑 2 = 𝑑 𝑥1 𝑖 + 𝑑 𝑦1 𝑗 + [(−𝑑 𝑥2 ) 𝑖 +(− 𝑑 𝑦2 ) 𝑗 ] 𝑑 1 − 𝑑 2 = 𝑑 𝑥1 − 𝑑 𝑥2 𝑖 + (𝑑 𝑦1 − 𝑑 𝑦2 ) 𝑗

Comprehension Check 2: (a) In unit-vector notation, what is the sum of 𝑎 + 𝑏 if 𝑎 = −12.0 𝑚 𝑖 +(3.0 𝑚) 𝑗 and 𝑏 = 8.0 𝑚 𝑖 +(7.0 𝑚) 𝑗 . (b) What is the magnitude of the 𝑎 + 𝑏 ? (c) What is the direction of 𝑎 + 𝑏 ?

Exit Ticket: Explain the difference between a scalar and a vector. Give an example of a scalar. Give an example of a vector. A ship sets out to sail to a point 120 km due north. An unexpected storm blows the ship to a point 100 km due east of its starting point. How far and in what direction must the ship now sail to reach its original destination? You are attempting to push a heavy box across the floor. You push the box with a force of 50 N at an angle of 35°. What are the x and y components of this force? Write this force vector in unit-vector notation.