Math 211: Worksheet 12.1-12.2 Day 2
Recall that a vector from the point (x1, y1, z1) to (x2, y2, z2) has components hx2โx1, y2โy1, z2โz1i.
Where the pointy brackets indicate that it is a vector and not just a point. This vector is equivalent
to a vector through the origin with the same components.
(1) Determine if the following coordinate systems are right-handed or left handed.
(The arrow indicates the positive direction of the axis.)
(2) Use the Pythagorean Theorem (more than once!) to compute the length of a vector in R3
beginning at the origin and ending at (a1, a2, a3).
(3) We will use the standard basis vectors i=h1,0,0i,j=h0,1,0i,k=h0,0,1i. Write the
vector hฯ, โe, 73ias a sum of multiples of the standard basis vectors.
(4) Cylinders:
(a) What is the equation of a circle in R2centered at (โ3,6) with radius 2?
(b) Imagine R2as the plane where z= 0 and draw a picture of the above circle in this
plane in R3. Letting zvary freely you now have a cylinder.
(c) In conclusion, the points in R3that satisfy the equation from (a) are the points on a
cylinder with center line {(โ3,6, z)}
(5) Spheres:
(a) A sphere can be defined as all points in R3equidistant from some fixed point.
(b) Use the distance formula to write an equation that describes all points in R3that are
distance 4 from the origin.
(c) Repeat (b) with the point (1,2,3) instead of the origin.
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