Vectors in Three Dimensions
Extend position, addition and scalar multiples to space, with distance and the standard unit vectors. Later topics in this sequence use this notation.
What a learner can do afterwards
- Write a vector in component form and find its magnitude in space
- Find the distance between two points and the midpoint of the segment
- Express a vector as a combination of i, j and k
- Normalize a non-zero vector by dividing each component by its magnitude
- Subtract the coordinates of two points to form the vector between them
1 · Read
In the plane a vector had two components, (a, b). In space it has three, written <a, b, c>. Find its magnitude by squaring each component, adding, and taking the square root.
For example, <4, 4, 7> has magnitude, or length, equal to the square root of 16 + 16 + 49, which is 9.
The distance between two points is the magnitude of their difference. For A(-2, 1, 4) and B(1, 5, 4), the difference is <3, 4, 0>, so the distance is 5.
A midpoint averages matching coordinates. For R(-3, 2, 6) and S(5, 8, 0), the y-coordinate is (2 + 8) divided by 2, which is 5.
To find a unit vector in the same direction, divide every component by the magnitude. The vector <5, 12, 0> has magnitude 13, so its unit vector is <5/13, 12/13, 0>.
To point the other way, first normalize the vector, then change every sign. The vector <1, 2, 2> has magnitude 3, so its opposite unit vector is <-1/3, -2/3, -2/3>.
The standard unit vectors i, j and k are axis helpers pointing 1 unit along x, y and z in turn. Each component multiplies its helper, so <4, -3, 2> is 4i - 3j + 2k.
Three components per vector, distance from differences, midpoints from averages, and i, j, k as shorthand.
The lesson and quiz items were written for this dataset by its content-upgrade pipeline. The lesson examples, the unit vector section and quiz item q2 were later revised.
Extra items x1 to x6, x8 and x9 were written for this dataset. Extra items x7 and x10 draw on problems 12.2.55 and 12.2.1 of the WeBWorK Open Problem Library set for Rogawski, Calculus: Early Transcendentals, second edition. The rewritten readings were written for this dataset. Other reading entries summarise pages from Khan Academy, each credited in the file.
2 · Watch
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Where it sits
Where this leads
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18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.