좌표 벡터의 연산과 크기 계산

문제

좌표평면 위의 두 벡터 $\vec{a} = (1, 2)$와 $\vec{b} = (-1, 3)$에 대하여, 벡터 $\vec{c} = 2\vec{a} + \vec{b}$의 크기는?

<rect x="0" y="0" width="400" height="300" class="background"/>

<!-- Coordinate System Setup -->
<!-- Scale: 1 unit = 30 pixels -->
<!-- Origin (0,0) in math coordinates maps to (100, 250) in SVG coordinates -->
<!-- Function tx(x_math) = 100 + x_math * 30 -->
<!-- Function ty(y_math) = 250 - y_math * 30 -->

<!-- Grid lines -->
<g class="grid-line">
    <!-- Vertical grid lines -->
    <line x1="70" y1="250" x2="70" y2="40"/> <!-- x = -1 -->
    <line x1="130" y1="250" x2="130" y2="40"/> <!-- x = 1 -->
    <line x1="160" y1="250" x2="160" y2="40"/> <!-- x = 2 -->
    <!-- Horizontal grid lines -->
    <line x1="70" y1="220" x2="160" y2="220"/> <!-- y = 1 -->
    <line x1="70" y1="190" x2="160" y2="190"/> <!-- y = 2 -->
    <line x1="70" y1="160" x2="160" y2="160"/> <!-- y = 3 -->
    <line x1="70" y1="130" x2="160" y2="130"/> <!-- y = 4 -->
    <line x1="70" y1="100" x2="160" y2="100"/> <!-- y = 5 -->
    <line x1="70" y1="70" x2="160" y2="70"/> <!-- y = 6 -->
    <line x1="70" y1="40" x2="160" y2="40"/> <!-- y = 7 -->
</g>

<!-- Axes -->
<line x1="30" y1="250" x2="200" y2="250" class="axis"/> <!-- X-axis -->
<line x1="100" y1="10" x2="100" y2="280" class="axis"/> <!-- Y-axis -->

<!-- Origin -->
<circle cx="100" cy="250" r="3" class="origin-dot"/>
<text x="90" y="255" class="label">O</text>

<!-- Axis Labels -->
<text x="195" y="245" class="label">x</text>
<text x="85" y="15" class="label">y</text>

<!-- Ticks and numbers on axes -->
<!-- X-axis -->
<line x1="130" y1="247" x2="130" y2="253" stroke="#1F2937" stroke-width="1"/>
<text x="130" y="265" class="label">1</text>
<line x1="160" y1="247" x2="160" y2="253" stroke="#1F2937" stroke-width="1"/>
<text x="160" y="265" class="label">2</text>
<line x1="70" y1="247" x2="70" y2="253" stroke="#1F2937" stroke-width="1"/>
<text x="70" y="265" class="label">-1</text>

<!-- Y-axis -->
<line x1="97" y1="220" x2="103" y2="220" stroke="#1F2937" stroke-width="1"/>
<text x="85" y="223" class="label">1</text>
<line x1="97" y1="190" x2="103" y2="190" stroke="#1F2937" stroke-width="1"/>
<text x="85" y="193" class="label">2</text>
<line x1="97" y1="160" x2="103" y2="160" stroke="#1F2937" stroke-width="1"/>
<text x="85" y="163" class="label">3</text>
<line x1="97" y1="130" x2="103" y2="130" stroke="#1F2937" stroke-width="1"/>
<text x="85" y="133" class="label">4</text>
<line x1="97" y1="100" x2="103" y2="100" stroke="#1F2937" stroke-width="1"/>
<text x="85" y="103" class="label">5</text>
<line x1="97" y1="70" x2="103" y2="70" stroke="#1F2937" stroke-width="1"/>
<text x="85" y="73" class="label">6</text>
<line x1="97" y1="40" x2="103" y2="40" stroke="#1F2937" stroke-width="1"/>
<text x="85" y="43" class="label">7</text>

<!-- Vectors -->
<!-- Vector a: (1,2) -->
<line x1="100" y1="250" x2="130" y2="190" class="vector-style vector-initial"/>
<text x="140" y="185" class="label vector-label">a</text>

<!-- Vector b: (-1,3) -->
<line x1="100" y1="250" x2="70" y2="160" class="vector-style vector-initial"/>
<text x="60" y="155" class="label vector-label">b</text>

<!-- Vector 2a: 2*(1,2) = (2,4) -->
<line x1="100" y1="250" x2="160" y2="130" class="vector-style vector-intermediate"/>
<text x="170" y="125" class="label vector-label">2a</text>

<!-- Parallelogram/Triangle for vector addition (2a + b = c) -->
<!-- Dashed vector representing b translated to the head of 2a -->
<!-- Head of 2a is (160, 130). Vector b is (-1,3) which means (-30, 90) in SVG relative coords. -->
<!-- So, (160-30, 130-90) = (130, 40) which is the head of vector c. -->
<line x1="160" y1="130" x2="130" y2="40" class="vector-style vector-translated-dashed"/>
<text x="145" y="95" class="label vector-label">b</text>

<!-- Vector c (resultant): 2a + b = (2,4) + (-1,3) = (1,7) -->
<line x1="100" y1="250" x2="130" y2="40" class="vector-style vector-resultant"/>
<text x="140" y="35" class="label vector-label">c</text>