Maths Olympiad Prep

Library / /309 of 520

Algebra Difficulty 5.7 AIME, harder Find the answer

## Problem Statement

Write the decomposition of vector xx in terms of vectors p,q,rp, q, r:

x={−15;5;6}x=\{-15 ; 5 ; 6\}

p={0;5;1}p=\{0 ; 5 ; 1\}

q={3;2;−1}q=\{3 ; 2 ;-1\}

r={−1;1;0}r=\{-1 ; 1 ; 0\}

A number or a short expression. Spacing and $ signs are ignored.

Solution

## Solution

The desired decomposition of vector xx is:

x=α⋅p+β⋅q+γ⋅rx=\alpha \cdot p+\beta \cdot q+\gamma \cdot r

Or in the form of a system:

{α⋅p1+β⋅q1+γ⋅r1=x1α⋅p2+β⋅q2+γ⋅r2=x2α⋅p3+β⋅q3+γ⋅r3=x3 \left\{\begin{array}{l} \alpha \cdot p_{1}+\beta \cdot q_{1}+\gamma \cdot r_{1}=x_{1} \\ \alpha \cdot p_{2}+\beta \cdot q_{2}+\gamma \cdot r_{2}=x_{2} \\ \alpha \cdot p_{3}+\beta \cdot q_{3}+\gamma \cdot r_{3}=x_{3} \end{array}\right.

We obtain:

{3β−γ=−155α+2β+γ=5α−β=6 \left\{\begin{array}{l} 3 \beta-\gamma=-15 \\ 5 \alpha+2 \beta+\gamma=5 \\ \alpha-\beta=6 \end{array}\right.

Add the third row multiplied by -5 to the second row:

{3β−γ=−157β+γ=−25α−β=6 \left\{\begin{array}{l} 3 \beta-\gamma=-15 \\ 7 \beta+\gamma=-25 \\ \alpha-\beta=6 \end{array}\right.

Add the first row to the second row:

{3β−γ=−1510β=−40α−β=6 \left\{\begin{array}{l} 3 \beta-\gamma=-15 \\ 10 \beta=-40 \\ \alpha-\beta=6 \end{array}\right.

{3β−γ=−15β=−4α−β=6{3⋅(−4)−γ=−15β=−4α−(−4)=6{−γ=−3β=−4α=2{γ=3β=−4α=2 \begin{aligned} & \left\{\begin{array}{l} 3 \beta-\gamma=-15 \\ \beta=-4 \\ \alpha-\beta=6 \end{array}\right. \\ & \left\{\begin{array}{l} 3 \cdot(-4)-\gamma=-15 \\ \beta=-4 \\ \alpha-(-4)=6 \end{array}\right. \\ & \left\{\begin{array}{l} -\gamma=-3 \\ \beta=-4 \\ \alpha=2 \end{array}\right. \\ & \left\{\begin{array}{l} \gamma=3 \\ \beta=-4 \\ \alpha=2 \end{array}\right. \end{aligned}

The desired decomposition:

x=2p−4q+3rx=2 p-4 q+3 r

## Problem Kuznetsov Analytic Geometry 2−212-21

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.