Unit 2: Vectors
Arrows that carry both size and direction, added, subtracted, and multiplied two different ways.
5 topics · 19 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Scalars Versus Vectors
- How it worksHow do you represent a vector's magnitude and direction together as a single arrow on a page?
- Worked exampleFind the magnitude and direction of a displacement vector with components 3 m east and 4 m north.
- Common mistakeWhy isn't distance traveled the same thing as the magnitude of a displacement vector?
- Going furtherWhat happens to a vector's components when you rotate the coordinate axes it's measured against?
Your turn1 of 2
A hiker walks 9 m east, then 9 m north. What is the magnitude of the resulting displacement, in meters?Vector Addition and Subtraction
- How it worksHow does the tip-to-tail method turn two separate vectors into their single sum?
- How it worksHow do you subtract one vector from another by adding its negative?
- Worked exampleFind the resultant of a 3 m displacement east combined with a 4 m displacement north.
- Common mistakeWhy doesn't adding two vectors ever just mean adding their magnitudes together?
Your turn1 of 2
A cyclist rides 11 km east, then turns and rides 10 km north. What is the magnitude of the resultant displacement, in kilometers?Unit Vectors and Components
- How it worksHow do the unit vectors i-hat, j-hat, and k-hat let you write any vector as a sum of components?
- Worked exampleFind the i-hat, j-hat components of a vector with magnitude 10 at 37 degrees above the x-axis.
- Going furtherWhat changes about a vector's i-hat and j-hat components once you switch to polar coordinates instead?
Your turn1 of 2
A vector has magnitude 15 and points 55 degrees above the positive x-axis. What is its i-hat (x) component?The Dot Product
- How it worksHow does the dot product of two vectors turn the angle between them into a single number?
- Worked exampleFind the dot product of vectors A = (2, 3, 0) and B = (4, -1, 2).
- Common mistakeWhy isn't the dot product of two vectors itself a vector?
- Going furtherWhat happens to the dot product of two vectors as the angle between them grows from 0 to 180 degrees?
Your turn1 of 2
Find the dot product of vector A = (2, 2, 5) and vector B = (-3, -5, -4).The Cross Product
- How it worksHow does the right-hand rule tell you the direction of a cross product before you calculate anything?
- Worked exampleFind the cross product of vectors A = (1, 0, 0) and B = (0, 1, 0).
- Common mistakeWhy doesn't the cross product of two vectors come out the same regardless of the order you multiply them in?
- Ties togetherHow does the cross product you use for torque connect to the one you'll later use for magnetic force?
Your turn1 of 2
Vector A lies in the xy-plane with components (2, 2, 0) and vector B lies in the xy-plane with components (5, -3, 0). What is the z-component of A cross B?