Difference between revisions of "2013 Mock AIME I Problems/Problem 8"
(Created page with "== Problem == Let <math>\textbf{u}=4\textbf{i}+3\textbf{j}</math> and <math>\textbf{v}</math> be two perpendicular vectors in the <math>x-y</math> plane. If there are <math>n<...") |
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== Solution == | == Solution == | ||
− | <math>\boxed{020}</math>. | + | |
+ | Let <math>\theta</math> be the angle that <math>u</math> makes with the positive <math>x</math>-axis. Now, rotate the plane clockwise by <math>\theta</math> such that <math>u</math> is on the new <math>x'</math>-axis, perpendicular to the <math>y'</math>-axis, on which <math>v</math> now lies, as seen in the diagram below. Rotations do not affect the magnitude of the vectors <math>r_i</math>, so our answer will be the same. | ||
+ | |||
+ | <asy> | ||
+ | |||
+ | import geometry; | ||
+ | |||
+ | // x' and y' Axes | ||
+ | draw((-8,0)--(8,0), arrow=Arrows); | ||
+ | label("$x'$", (9,0)); | ||
+ | draw((0,-8)--(0,8), arrow=Arrows); | ||
+ | label("$y'$", (0,9)); | ||
+ | |||
+ | // Vectors | ||
+ | draw((0,0)--(5,0), red+linewidth(1.5), arrow=Arrow(TeXHead)); | ||
+ | label("$u$", midpoint((0,0)--(5,0)), S); | ||
+ | draw((0,0)--(0,6), blue+linewidth(1.5), arrow=Arrow(TeXHead)); | ||
+ | label("$v$", midpoint((0,0)--(0,6)), W); | ||
+ | |||
+ | // Right Angle Mark | ||
+ | markscalefactor = 0.12; | ||
+ | draw(rightanglemark((1,0),(0,0),(0,1))); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | Now, from the information given by the problem, it is clear that the rotated vectors <math>r_i'</math> are those with coordinates <math>(\pm1, \pm2)</math>. Thus, there are <math>4</math> of these vectors, each with a squared magnitude of <math>(\pm 1)^2+(\pm 2)^2 = 1+4 = 5</math>. Thus, our answer is <math>4 \cdot 5 = \boxed{020}</math>. | ||
== See also == | == See also == |
Latest revision as of 14:20, 30 July 2024
Problem
Let and
be two perpendicular vectors in the
plane. If there are
vectors
for
in the same plane having projections of
and
along
and
respectively, then find
(Note:
and
are unit vectors such that
and
, and the projection of a vector
onto
is the length of the vector that is formed by the origin and the foot of the perpendicular of
onto
.)
Solution
Let be the angle that
makes with the positive
-axis. Now, rotate the plane clockwise by
such that
is on the new
-axis, perpendicular to the
-axis, on which
now lies, as seen in the diagram below. Rotations do not affect the magnitude of the vectors
, so our answer will be the same.
Now, from the information given by the problem, it is clear that the rotated vectors are those with coordinates
. Thus, there are
of these vectors, each with a squared magnitude of
. Thus, our answer is
.