Unit 9: Vectors
Quantities with both size and direction, added tip to tail, broken into components, and combined with the dot product to measure the angle between them.
6 topics · 23 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.
Vectors, Magnitude, and Direction
- How it worksHow do a vector's magnitude and direction combine into an arrow that represents both facts at once?
- Worked exampleFind the magnitude and direction angle of the vector with initial point (1,2) and terminal point (4,6).
- Common mistakeWhy isn't a vector's magnitude the same thing as either of its components?
Your turn1 of 2
Find the magnitude of the vector from (3,3) to (x, 12), as a function of x.Vector Components and Unit Vectors
- How it worksHow do the component form <a,b> and the i, j unit vector form describe the exact same vector?
- Worked exampleFind the component form and the i, j form of a vector with magnitude 10 at an angle of 60 degrees.
- Common mistakeWhy doesn't a unit vector have to point along an axis, since it only has to have length 1?
- Ties togetherHow does writing a vector in component form make adding two vectors as easy as adding numbers?
Your turn1 of 2
Write the x-component of a vector as a function of its direction angle theta (in radians), given magnitude 14.Vector Addition and Scalar Multiplication
- How it worksHow does placing vectors tip to tail show their sum as the arrow from the first start to the last end?
- Worked exampleFind the sum of vectors <3,4> and <-1,2>, and find 3 times the vector <2,-1>.
- Worked exampleFind the vector -2 times <3,-5> plus the vector <1,4>.
- Common mistakeWhy doesn't adding two vectors' magnitudes give you the magnitude of their sum?
Your turn1 of 2
Find the x-component of the sum of vectors <4,3> and <8,-4>.The Dot Product
- How it worksHow does the dot product formula a1·b1 + a2·b2 relate to the angle between two vectors?
- Worked exampleFind the dot product of <2,3> and <-1,4>, and use it to find the angle between them.
- Common mistakeWhy does a dot product of zero mean two vectors are perpendicular, not that one of them is zero?
- Common mistakeWhy isn't the dot product of two vectors itself a vector, the way vector addition is?
Your turn1 of 2
Find the dot product of <x, 4> and <3, 8>, as a function of x.Vector Projections
- How it worksHow does the dot product build the formula for projecting one vector onto another?
- Worked exampleFind the projection of vector <3,4> onto vector <1,0>.
- Common mistakeWhy isn't the projection of vector a onto vector b the same as the projection of b onto a?
- Ties togetherHow does the dot product formula turn directly into the vector projection formula?
Your turn1 of 2
Find the scalar projection of vector <x, 4> onto vector <6, 8>, as a function of x.Vectors in Applications
- How it worksHow do you add a velocity vector and a wind vector to find a plane's true speed and direction over the ground?
- Worked exampleFind the resultant speed and direction of a plane flying at 300 mph due north through a 50 mph wind blowing from the west.
- Worked exampleFind the tension in two ropes holding a 100 lb weight when the ropes make angles of 30 and 45 degrees with the ceiling.
- Going furtherWhat happens to the resultant vector's direction as a crosswind's speed increases relative to the plane's own speed?
Your turn1 of 2
A plane flies at 309 mph due north through a crosswind of speed w mph blowing from the west. Write the resultant ground speed as a function of w.