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Quizz 1 (Notes, books, and calculators are not authorized)
Show all your work in the blank space you are given on the exam sheet. Always justify your
answer. Answers with no justification will not be graded.
Question 1: Find a parametric representation of the line in R4 that passes through P =
(1, 1, 3, 3) and is orthogonal to the hyperplane 2x1 + 4x2 + 6x3 − 8x4 = 5.
A normal vector to the hyperplane 2x1 + 4x2 + 6x3 − 8x4 = 5 is u := (2, 4, 6, −8). The line in
question is the collection of points X = (x1 , x2 , x3 , x4 ) so that X − P is parallel to u. This means
that there is a real number t so that
X − P = tu = (2t, 4t, 6t, −8t).
The parametric representation of the line is
x1 = 1 + 2t,
x2 = 1 + 4t,
x3 = 3 + 6t,
x4 = 3 − 8t.
Question 2: Find a unit vector orthogonal to u = (1, 2, 3) and v = (1, −1, 2).
It is a known property of the cross product that the following vector w := u × v is orthogonal to u
and v. By definition
w = (2 × 2 − 3 × (−1), 3 × 1 − 2 × 1, 1 × (−1) − 2 × 1) = (7, 1, −3).
By normalizing w we obtain the desired vector:
1
w
= √ (7, 1, −3).
kwk
59
(1)
2
Quizz 1, September 7, 2010
Question 3: Find an equation of the hyperplane in R4 that passes through P = (3, −4, 1, −2)
and is normal to u = (1, 1, −1, −1).
By definition H is the collection of points X = (x1 , x2 , x3 , x4 ) so that (X − P )·u. This means
0 =(X − P )·(1, 1, −1, −1) = (x1 − 3, x2 + 4, x3 − 1, x4 + 2)·(1, 1, −1, −1)
= x1 − 3 + x2 + 4 − (x3 − 1) − (x4 + 2) = x1 + x2 − x3 − x4 .
The equation of the hyperplane in question
0 = x1 + x2 − x3 − x4 .
Question 4: Prove the triangle inequality: ku + vk ≤ kuk + kvk.
By definition of the dot product, we have the folowing equality:
ku + vk2 := (u + v)2 = (u + v)·(u + v) = u·u + u·v + v·u + v·u.
Using the Cauchy-Schwarz inequality, we obtain
ku + vk2 ≤ kuk2 + kukkvk + kvkkuk + kvk2 = (kuk + kvk)2 .
Taking the square root gives the triangle inequality:
ku + vk ≤ kuk + kvk.