MATH 2242: Graded Homework # 1 - DUE 1/22/09
1. Let a= (1,1), b= (2,−1), α=−2, β= 3. Calculate:
(a) αa+βbalgebraically and geometrically.
(b) Describe in words the vector b−a.
(c) Normalize the vector b(i.e find the vector ˆ
b).
(d) Find the orthogonal projection of aalong b.
2. Let Lbe the line containing the points (1,1,1) and (2,2,3). Let L0be
the line with direction vector v= (1,−1,−1) and containing the point
(0,2,2).
(a) Find an equation L(t) for the line L.
(b) Find an equation L0(t) for the line L0.
(c) Do Land L0intersect? (Hint: Can you find a value of tfor which
L(t) = L0(t)?)
3. (a) Find the equation of the plane Pwith normal vector n= (−1,4,2)
and containing the point p0= (1,1,1).
(b) For what value of zis the point p= (0,1, z) in the plane P?
4. Let a= (−1,0,2), b= (3,4,5), c= (1,1,1), α= 2, and β=−3.
Calculate the following:
(a) a·b. Are aand borthogonal (i.e. perpendicular)? Explain.
(b) a×band b×a(Hint: for the second one, use a property of cross
products - don’t rework it).
(c) a×(b×c) (Hint: Use the BAC-CAB rule).
(d) a·(b×c) (Hint: Use the scalar triple product rule).
5. Recall from physics that the work Wdone by a force Facting over a
displacement dis given by W=F·d. Suppose an electric field exerts
the constant force F= 2ˆ
i−3ˆ
j+ˆ
kwhile moving a charged particle from
the point a= (−1,3,2) to the point b= (0,7,3). Find the work W
done by the field on the particle. (Note: Normally all physics equations
carry units – but nevermind that in this class.)
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